Chapter 12Resource Masters
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ISBN: 0-07-869139-7 Advanced Mathematical ConceptsChapter 12 Resource Masters
1 2 3 4 5 6 7 8 9 10 XXX 11 10 09 08 07 06 05 04
© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts
Vocabulary Builder . . . . . . . . . . . . . . . . vii-ix
Lesson 12-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 509Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Lesson 12-2Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 512Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 514
Lesson 12-3Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 515Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Lesson 12-4Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 518Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 520
Lesson 12-5Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 521Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 523
Lesson 12-6Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 524Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 526
Lesson 12-7Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 527Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 529
Lesson 12-8Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 530Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Lesson 12-9Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 533Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 535
Chapter 12 AssessmentChapter 12 Test, Form 1A . . . . . . . . . . . 537-538Chapter 12 Test, Form 1B . . . . . . . . . . . 539-540Chapter 12 Test, Form 1C . . . . . . . . . . . 541-542Chapter 12 Test, Form 2A . . . . . . . . . . . 543-544Chapter 12 Test, Form 2B . . . . . . . . . . . 545-546Chapter 12 Test, Form 2C . . . . . . . . . . . 547-548Chapter 12 Extended Response
Assessment . . . . . . . . . . . . . . . . . . . . . . . 549Chapter 12 Mid-Chapter Test . . . . . . . . . . . . 550Chapter 12 Quizzes A & B . . . . . . . . . . . . . . 551Chapter 12 Quizzes C & D. . . . . . . . . . . . . . 552Chapter 12 SAT and ACT Practice . . . . 553-554Chapter 12 Cumulative Review . . . . . . . . . . 555Trigonometry Semester Test . . . . . . . . . 557-561Trigonometry Final Test . . . . . . . . . . . . 563-570
SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1
SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A23
Contents
© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts
A Teacher’s Guide to Using theChapter 12 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 12 Resource Masters include the corematerials needed for Chapter 12. These materials include worksheets, extensions,and assessment options. The answers for these pages appear at the back of thisbooklet.
All of the materials found in this booklet are included for viewing and printing inthe Advanced Mathematical Concepts TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-ix include a student study tool that presents the key vocabulary terms from the chapter. Students areto record definitions and/or examples for eachterm. You may suggest that students highlight orstar the terms with which they are not familiar.
When to Use Give these pages to studentsbefore beginning Lesson 12-1. Remind them toadd definitions and examples as they completeeach lesson.
Study Guide There is one Study Guide master for each lesson.
When to Use Use these masters as reteachingactivities for students who need additional reinforcement. These pages can also be used inconjunction with the Student Edition as aninstructional tool for those students who havebeen absent.
Practice There is one master for each lesson.These problems more closely follow the structure of the Practice section of the StudentEdition exercises. These exercises are of averagedifficulty.
When to Use These provide additional practice options or may be used as homeworkfor second day teaching of the lesson.
Enrichment There is one master for eachlesson. These activities may extend the conceptsin the lesson, offer a historical or multiculturallook at the concepts, or widen students’perspectives on the mathematics they are learning. These are not written exclusively forhonors students, but are accessible for use withall levels of students.
When to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.
© Glencoe/McGraw-Hill v Advanced Mathematical Concepts
Assessment Options
The assessment section of the Chapter 12Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessments
Chapter Tests• Forms 1A, 1B, and 1C Form 1 tests contain
multiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with average-level students, and Form 1C is intended foruse with basic-level students. These tests aresimilar in format to offer comparable testingsituations.
• Forms 2A, 2B, and 2C Form 2 tests arecomposed of free-response questions. Form2A is intended for use with honors-levelstudents, Form 2B is intended for use withaverage-level students, and Form 2C isintended for use with basic-level students.These tests are similar in format to offercomparable testing situations.
All of the above tests include a challengingBonus question.
• The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided for assessment.
Intermediate Assessment• A Mid-Chapter Test provides an option to
assess the first half of the chapter. It iscomposed of free-response questions.
• Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter.
Continuing Assessment• The SAT and ACT Practice offers
continuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitative-comparison, and grid-in questions. Bubble-in and grid-in answer sections are providedon the master.
• The Cumulative Review provides studentsan opportunity to reinforce and retain skillsas they proceed through their study ofadvanced mathematics. It can also be usedas a test. The master includes free-responsequestions.
Answers• Page A1 is an answer sheet for the SAT and
ACT Practice questions that appear in theStudent Edition on page 835. Page A2 is ananswer sheet for the SAT and ACT Practicemaster. These improve students’ familiaritywith the answer formats they may encounterin test taking.
• The answers for the lesson-by-lesson masters are provided as reduced pages withanswers appearing in red.
• Full-size answer keys are provided for theassessment options in this booklet.
© Glencoe/McGraw-Hill vi Advanced Mathematical Concepts
Chapter 12 Leveled Worksheets
Glencoe’s leveled worksheets are helpful for meeting the needs of everystudent in a variety of ways. These worksheets, many of which are foundin the FAST FILE Chapter Resource Masters, are shown in the chartbelow.
• Study Guide masters provide worked-out examples as well as practiceproblems.
• Each chapter’s Vocabulary Builder master provides students theopportunity to write out key concepts and definitions in their ownwords.
• Practice masters provide average-level problems for students who are moving at a regular pace.
• Enrichment masters offer students the opportunity to extend theirlearning.
primarily skillsprimarily conceptsprimarily applications
BASIC AVERAGE ADVANCED
Study Guide
Vocabulary Builder
Parent and Student Study Guide (online)
Practice
Enrichment
4
5
3
2
Five Different Options to Meet the Needs of Every Student in a Variety of Ways
1
© Glencoe/McGraw-Hill vii Advanced Mathematical Concepts
Reading to Learn MathematicsVocabulary Builder
NAME _____________________________ DATE _______________ PERIOD ________
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 12.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.
Vocabulary Term Foundon Page Definition/Description/Example
arithmetic mean
arithmetic sequence
arithmetic series
Binomial Theorem
common difference
common ratio
comparison test
convergent series
divergent series
escaping point
Chapter
12
(continued on the next page)
© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts
Reading to Learn MathematicsVocabulary Builder (continued)
NAME _____________________________ DATE _______________ PERIOD ________
Vocabulary Term Foundon Page Definition/Description/Example
Euler’s Formula
exponential series
Fibonacci sequence
fractal geometry
geometric mean
geometric sequence
geometric series
index of summation
infinite sequence
infinite series
limit
Chapter
12
(continued on the next page)
© Glencoe/McGraw-Hill ix Advanced Mathematical Concepts
Reading to Learn MathematicsVocabulary Builder (continued)
NAME _____________________________ DATE _______________ PERIOD ________
Vocabulary Term Foundon Page Definition/Description/Example
mathematical induction
n factorial
nth partial sum
orbit
Pascal’s Triangle
prisoner point
ratio test
recursive formula
sequence
sigma notation
term
trigonometric series
Chapter
12
BLANK
© Glencoe/McGraw-Hill 509 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
12-1
Arithmetic Sequences and SeriesA sequence is a function whose domain is the set of naturalnumbers. The terms of a sequence are the range elements of thefunction. The difference between successive terms of an arithmeticsequence is a constant called the common difference, denoted as d. An arithmetic series is the indicated sum of the terms of anarithmetic sequence.
Example 1 a. Find the next four terms in the arithmeticsequence �7, �5, �3, . . . .
b. Find the 38th term of this sequence.
a. Find the common difference.a2 � a1 � �5 � (�7) or 2
The common difference is 2. Add 2 to the thirdterm to get the fourth term, and so on.a4 � �3 � 2 or �1 a5 � �1 � 2 or 1a6 � 1 � 2 or 3 a7 � 3 � 2 or 5
The next four terms are �1, 1, 3, and 5.
b. Use the formula for the nth term of anarithmetic sequence.an � a1 � (n � 1)da38 � �7 � (38 �1)2 n � 38, a1 � �7, d � 2a38 � 67
Example 2 Write an arithmetic sequence that has threearithmetic means between 3.2 and 4.4.
The sequence will have the form 3.2, ? , ? , ? , 4.4.
First, find the common difference.an � a1 � (n � 1)d4.4 � 3.2 � (5 � 1)d n � 5, a5 � 4.4, a1 � 3.24.4 � 3.2 � 4dd � 0.3
Then, determine the arithmetic means.
The sequence is 3.2, 3.5, 3.8, 4.1, 4.4.
Example 3 Find the sum of the first 50 terms in the series11 � 14 � 17 � . . . � 158.
Sn � �n2�(a1 � an)
S50 � �520�(11 � 158) n � 50, a1 � 11, a50 � 158
� 4225
a2 a3 a4
3.2 � 0.3 � 3.5 3.5 � 0.3 � 3.8 3.8 � 0.3 � 4.1
© Glencoe/McGraw-Hill 510 Advanced Mathematical Concepts
Arithmetic Sequences and SeriesFind the next four terms in each arithmetic sequence.
1. �1.1, 0.6, 2.3, . . . 2. 16, 13, 10, . . . 3. p, p � 2, p � 4, . . .
For exercises 4–12, assume that each sequence or series is arithmetic.
4. Find the 24th term in the sequence for which a1 � �27 and d � 3.
5. Find n for the sequence for which an � 27, a1 � �12, and d � 3.
6. Find d for the sequence for which a1 � �12 and a23 � 32.
7. What is the first term in the sequence for which d � �3 and a6 � 5?
8. What is the first term in the sequence for which d � ��13� and a7 � �3?
9. Find the 6th term in the sequence �3 � �2�, 0, 3 � �2�, . . . .
10. Find the 45th term in the sequence �17, �11, �5, . . . .
11. Write a sequence that has three arithmetic means between 35 and 45.
12. Write a sequence that has two arithmetic means between �7 and 2.75.
13. Find the sum of the first 13 terms in the series �5 � 1 � 7 � . . . � 67.
14. Find the sum of the first 62 terms in the series �23 � 21.5 � 20 � . . . .
15. Auditorium Design Wakefield Auditorium has 26 rows, and the first row has 22seats. The number of seats in each row increases by 4 as you move toward the backof the auditorium. What is the seating capacity of this auditorium?
PracticeNAME _____________________________ DATE _______________ PERIOD ________
12-1
© Glencoe/McGraw-Hill 511 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
12-1
Quadratic Formulas for SequencesAn ordinary arithmetic sequence is formed using a rule such as bn � c. The first term is c, b is called the common difference, and ntakes on the values 0, 1, 2, 3, and so on. The value of term n � 1equals b(n � 1) � c or bn � b � c. So, the value of a term is a function of the term number.
Some sequences use quadratic functions. A method called finitedifferences can be used to find the values of the terms. Notice whathappens when you subtract twice as shown in this table.
n an2 � bn � c0 c
a � b1 a � b � c
3a � b2a
2 4a � 2b � c5a � b
2a3 9a � 3b � c
7a � b2a
4 16a � 4b � cA sequence that yields a common difference after two subtractionscan be generated by a quadratic expression. For example, thesequence 1, 5, 12, 22, 35, . . . gives a common difference of 3 after twosubtractions. Using the table above, you write and solve three equa-tions to find the general rule. The equations are 1 � c,5 � a � b � c, and 12 � 4a � 2b � c.Solve each problem.
1. Refer to the sequence in the example above. Solve the system ofequations for a, b, and c and then find the quadratic expressionfor the sequence. Then write the next three terms.
2. The number of line segments connecting n points forms thesequence 0, 0, 1, 3, 6, 10, . . . , in which n is the number of pointsand the term value is the number of line segments. What is thecommon difference after the second subtraction? Find a quadraticexpression for the term value.
3. The maximum number of regions formed by n chords in a circleforms the sequence 1, 2, 4, 7, 11, 16, . . . (A chord is a line segmentjoining any two points on a circle.) Draw circles to illustrate thefirst four terms of the sequence. Then find a quadratic expressionfor the term value.
© Glencoe/McGraw-Hill 512 Advanced Mathematical Concepts
Geometric Sequences and SeriesA geometric sequence is a sequence in which each term after thefirst, a1, is the product of the preceding term and the commonratio, r. The terms between two nonconsecutive terms of ageometric sequence are called geometric means. The indicatedsum of the terms of a geometric sequence is a geometric series.Example 1 Find the 7th term of the geometric sequence
157, �47.1, 14.13, . . . .
First, find the common ratio.a2 � a1 � �47.1 � 157 or �0.3The common ratio is �0.3.
Then, use the formula for the nth term of ageometric sequence.an � a1r
n � 1
a7 � 157(�0.3)6 n � 7, a1 � 157, r � �0.3a7 � 0.114453
The 7th term is 0.114453.
Example 2 Write a sequence that has two geometric meansbetween 6 and 162.The sequence will have the form 6, ? , ? , 162.
First, find the common ratio.an � a1r
n � 1
162 � 6r3 a4 � 162, a1 � 6, n � 427 � r3 Divide each side by 6.3 � r Take the cube root of each side.
Then, determine the geometric sequence.a2 � 6 � 3 or 18 a3 � 18 � 3 or 54The sequence is 6, 18, 54, 162.
Example 3 Find the sum of the first twelve terms of thegeometric series 12 � 12�2� � 24 � 24�2� � . . . .
First, find the common ratio.a2 � a1 � � 12�2� � 12 or � �2�The common ratio is � �2�.
Sn � �a1
1�
�
ar1r
n
�
S12 ��12
1
�
�
1
(2
�
(�
�
�
2�
2�
))12
� n � 12, a1 � 12, r � ��2�
S12 � 756(1 ��2�) Simplify.
The sum of the first twelve terms of the series is 756(1 ��2�).
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
12-2
© Glencoe/McGraw-Hill 513 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
Geometric Sequences and Series
Determine the common ratio and find the next three terms of each geometricsequence.
1. �1, 2, �4, . . . 2. �4, �3, ��94�, . . . 3. 12, �18, 27, . . .
For exercises 4–9, assume that each sequence or series is geometric.
4. Find the fifth term of the sequence 20, 0.2, 0.002, . . . .
5. Find the ninth term of the sequence �3�, �3, 3�3�, . . . .
6. If r � 2 and a4 � 28, find the first term of the sequence.
7. Find the first three terms of the sequence for which a4 � 8.4 and r � 4.
8. Find the first three terms of the sequence for which a6 � �312� and r � �12�.
9. Write a sequence that has two geometric means between 2 and 0.25.
10. Write a sequence that has three geometric means between �32 and �2.
11. Find the sum of the first eight terms of the series �34� � �290� � �1
2070� � . . . .
12. Find the sum of the first 10 terms of the series �3 � 12 � 48 � . . . .
13. Population Growth A city of 100,000 people is growing at a rate of 5.2% per year. Assuming this growth rate remains constant, estimate the population of the city 5 years from now.
12-2
© Glencoe/McGraw-Hill 514 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
12-2
Sequences as FunctionsA geometric sequence can be defined as a function whose domain is the set of positive integers.
n � 1 2 3 4 ...↓ ↓ ↓ ↓ ↓
f (n) � ar1 – 1 ar2 – 1 ar3 – 1 ar4 – 1 ...In the exercises, you will have the opportunity to explore geometricsequences from a function and graphing point of view.
Graph each geometric sequence for n = 1, 2, 3 and 4.
1. f (n) � 2n 2. f (n) � (0.5)n
3. f (n) � (–2)n 4. f (n) � (–0.5)n
5. Describe how the graph of a geometric sequence depends on thecommon ratio.
6. Let f (n) � 2n, where n is a positive integer.a. Show graphically that for any M the graph of f (n) rises
above and stays above the horizontal line y � M.b. Show algebraically that for any M, there is a positive
integer N such that 2n � M for all n � N.
© Glencoe/McGraw-Hill 515 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
Infinite Sequences and SeriesAn infinite sequence is one that has infinitely many terms. Aninfinite series is the indicated sum of the terms of an infinitesequence.
Example 1 Find lim �4n2
n�2 �
n1� 3�.
Divide each term in the numerator and thedenominator by the highest power of n to produce anequivalent expression. In this case, n2 is the highestpower.
lim �4n2
n�2 �
n1� 3�� lim
� lim Simplify.
� Apply limit theorems.
��4 �10
��
03 � 0� or 4 lim 4 � 4, lim �n
1� � 0, lim 3 � 3,
lim �n1
2� � 0, lim 1 � 1Thus, the limit is 4.
Example 2 Find the sum of the series �32� � �38� � �332� � . . . .
In the series a1 � �32� and r � ��14�.
Since �r � < 1, S � �1a�
1
r�.
S � �1a�
1r� � a1 � �32� and r � ��4
1�
� �1120� or 1�5
1�
The sum of the series is 1�15�.
�32���1 � ���1
4��
lim 4 � lim �n1� � lim 3 � lim �n
12�
���lim 1 � lim �
n1
2�
4 � �n1� � �n
32�
��1 � �
n1
2�
�4nn2
2� � �n
n2� � �n
32�
���nn
2
2� � �n1
2�
12-3
n→∞
n→∞ n→∞ n→∞
n→∞n→∞
n→∞
n→∞
n→∞ n→∞ n→∞
n→∞ n→∞
© Glencoe/McGraw-Hill 516 Advanced Mathematical Concepts
Infinite Sequence and SeriesFind each limit, or state that the limit does not exist andexplain your reasoning.
1. limn→∞
�nn
2
2
��
11� 2. lim
n→∞�43
nn2
2��
54n�
3. limn→∞
�5n62
n� 1� 4. lim
n→∞�(n � 1
5)(n3
2n � 1)�
5. limn→∞
�3n �
4n(�
21)n
� 6. limn→∞
�n3
n�2
1�
Write each repeating decimal as a fraction.
7. 0.7�5� 8. 0.5�9�2�
Find the sum of each infinite series, or state that the sum does not exist and explain your reasoning.
9. �25� � �265� � �1
1285� � . . . 10. �34� � �18
5� � �7156� � . . .
11. Physics A tennis ball is dropped from a height of 55 feet andbounces �35� of the distance after each fall.a. Find the first seven terms of the infinite series representing
the vertical distances traveled by the ball.
b. What is the total vertical distance the ball travels before coming to rest?
PracticeNAME _____________________________ DATE _______________ PERIOD ________
12-3
© Glencoe/McGraw-Hill 517 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
12-3
Solving Equations Using SequencesYou can use sequences to solve many equations. For example,consider x2 � x – 1 � 0. You can proceed as follows.
x2 � x – 1 � 0
x(x � 1) � 1
x �
Next, define the sequence: a1 � 0 and an � .
The limit of the sequence is a solution to the original equation.
1. Let a1 � 0 and an � .
a. Write the first five terms of the sequence. Do not simplify.
b. Write decimals for the first five terms of the sequence.
c. Use a calculator to compute a6, a7, a8, and a9. Compare a9 withthe positive solution of x2 � x – 1 � 0 found by using the quadratic formula.
2. Use the method described above to find a root of 3x2 – 2x – 3 � 0.
3. Write a BASIC program using the procedure outlined above to find a root of the equation 3x2 – 2x – 3 � 0. In the program,
let a1 � 0 and an � . Run the program. Compare the
time it takes to run the program to the time it takes to evaluatethe terms of the sequence by using a calculator.
3��3an�1�2
1��1 � an � 1
1��1 � an � 1
1�1 � x
© Glencoe/McGraw-Hill 518 Advanced Mathematical Concepts
Convergent and Divergent SeriesIf an infinite series has a sum, or limit, the series is convergent. Ifa series is not convergent, it is divergent. When a series is neitherarithmetic nor geometric and all the terms are positive, you can usethe ratio test or the comparison test to determine whether theseries is convergent or divergent.
Example 1 Use the ratio test to determine whether the series �12
�12� � �22
�23� � �32
�34� � �42
�45� � . . . is convergent or divergent.
First, find the nth term. Then use the ratio test.
an � �n(n
2�n
1)� an � 1 � �
(n �21n)�
(n1� 2)
�
r � lim
r � lim �(n �21n)�
(n1� 2)
� � �n(n2�
n
1)� Multiply by the reciprocal of the divisor.
r � lim �n2�n
2� �2n2
�
n
1� � �12�
r � lim Divide by the highest power of n and apply limit theorems.
r � �12� Since r � 1, the series is convergent.
Example 2 Use the comparison test to determine whether the series
�41
2� � �
71
2� � �
1102� � �
1132� � . . . is convergent or divergent.
The general term of the series is �(3n �
11)2
�. The general
term of the convergent series 1 � �21
2� � �
31
2� � �
41
2� � . . .
is �n1
2�. Since �
(3n �
11)2
� � �n12
� for all n 1, the series
�41
2� � �
71
2� � �
1102� � �
1132� � . . . is also convergent.
1 � �n2�
�2
�(n �
21n)�
(n1� 2)
���
�n(n
2�n
1)�
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
12-4
Let an and an � 1 represent two consecutive terms of a series of positive terms.
Ratio Test Suppose lim �an
a�
n
1� exists and r � lim �
an
a�
n
1�. The series is convergent if r � 1 and
divergent if r � 1. If r � 1, the test provides no information.
• A series of positive terms is convergent if, for n � 1, each term of the series is equal to or less than the value of the corresponding term of some
Comparison convergent series of positive terms.Test • A series of positive terms is divergent if, for n � 1, each term of the series is
equal to or greater than the value of the corresponding term of some divergent series of positive terms.
n→∞
n→∞
n→∞
n→∞
n→∞ n→∞
© Glencoe/McGraw-Hill 519 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
Convergent and Divergent Series
Use the ratio test to determine whether each series isconvergent or divergent.
1. �12
� � �22
2
2� � �3
2
2
3� � �4
2
2
4� � . . . 2. 0.006 � 0.06 � 0.6 � . . .
3. �1 �
42 � 3� � �
1 � 28� 3 � 4� � �
1 � 2 �
136
� 4 � 5� � . . .
4. 5 � �353� � �
553�� �
753� � . . .
Use the comparison test to determine whether each series isconvergent or divergent.
5. 2 � �223� � �
323� � �
423� � . . . 6. �52� � 1 � �58� � �1
51� � . . .
7. Ecology A landfill is leaking a toxic chemical. Six months afterthe leak was detected, the chemical had spread 1250 meters fromthe landfill. After one year, the chemical had spread 500 metersmore, and by the end of 18 months, it had reached an additional200 meters.a. If this pattern continues, how far will the chemical spread
from the landfill after 3 years?
b. Will the chemical ever reach the grounds of a hospital located2500 meters away from the landfill? Explain.
12-4
© Glencoe/McGraw-Hill 520 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
12-4
Alternating SeriesThe series below is called an alternating series.
1 – 1 � 1 – 1 � ...
The reason is that the signs of the terms alternate. An interestingquestion is whether the series converges. In the exercises, you willhave an opportunity to explore this series and others like it.
1. Consider 1 – 1 � 1 – 1 � ... .a. Write an argument that suggests that the sum is 1.
b. Write an argument that suggests that the sum is 0.
c. Write an argument that suggests that there is no sum.(Hint: Consider the sequence of partial sums.)
If the series formed by taking the absolute values of the terms of agiven series is convergent, then the given series is said to beabsolutely convergent. It can be shown that any absolutely convergent series is convergent.
2. Make up an alternating series, other than a geometric series withnegative common ratio, that has a sum. Justify your answer.
© Glencoe/McGraw-Hill 521 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
12-5
Sigma Notation and the nth TermA series may be written using sigma notation.
maximum value of n →�an ← expression for general term
starting value of n →
index of summation
Example 1 Write each expression in expanded formand then find the sum.
a. � (n � 2)
First, write the expression in expanded form.
� (n � 2) � (1 � 2) � (2 � 2) � (3 � 2) � (4 � 2) � (5 � 2)
Then, find the sum by simplifying the expandedform. 3 � 4 � 5 � 6 � 7 � 25
b. � 2��14��m
� 2��14��m
� 2��14��1� 2��14��2
� 2��14��3� . . .
� �12� � �18� � �312� � . . .
This is an infinite series. Use the formula S � �1a�
1
r�.
S � a1 � �12�, r � �14�
S � �32�
Example 2 Express the series 26 � 37 � 50 � 65 � . . . � 170using sigma notation.
Notice that each term is one more than a perfectsquare. Thus, the nth term of the series is n2 � 1.Since 52 � 1 � 26 and 132 � 1 � 170, the index ofsummation goes from n � 5 to n � 13.
Therefore, 26 � 37 � 50 � 65 �. . . � 170 � � (n2 � 1).
�12
�
�1 � �14�
k
n�1
5
n�1
5
n�1
∞
m�1∞
m�1
13
n�5
↑
© Glencoe/McGraw-Hill 522 Advanced Mathematical Concepts
Sigma Notation and the nth TermWrite each expression in expanded form and then find the sum.
1. �5
n�3(n2 � 2n) 2. �
4
q�1�q2�
3. �5
t�1t(t � 1) 4. �
3
t�0(2t � 3)
5. �5
c�2(c � 2)2 6. �
∞
i�110��12��
i
Express each series using sigma notation.
7. 3 � 6 � 9 � 12 � 15 8. 6 � 24 � 120 � . . . � 40,320
9. �11� � �14� � �19� � . . . � �1010� 10. 24 � 19 � 14 � . . . � (�1)
11. Savings Kathryn started saving quarters in a jar. She beganby putting two quarters in the jar the first day and then sheincreased the number of quarters she put in the jar by one additional quarter each successive day.a. Use sigma notation to represent the total number of quarters
Kathryn had after 30 days.
b. Find the sum represented in part a.
PracticeNAME _____________________________ DATE _______________ PERIOD ________
12-5
© Glencoe/McGraw-Hill 523 Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Street Networks: Finding All Possible RoutesA section of a city is laid out in square blocks. Going north from the intersection of 1st Avenue and 1st Street, the avenues are 1st, 2nd, 3rd, and so on. Going east, the streets are numbered in the same way.
Factorials can be used to find the number,r(e, n), of different routes between two intersections.
The number of streets going east is e; the number of avenues goingnorth is n.
The following problems examine the possible routes from one location to another. Assume that you never use a route that is unnecessarily long. Assume that e 1 and n 1.
Solve each problem.
1. List all the possible routes from 1st Street and 1st Avenue to 4th Street and 3rd Avenue. Use ordered pairs to show the routes,with street numbers first and avenue numbers second. Each route must start at (1, 1) and end at (4, 3).
2. Use the formula to compute the number of routes from (1, 1) to (4, 3).There are 4 streets going east and 3 avenues going north.
3. Find the number of routes from 1st Street and 1st Avenue to 7th Street and 6th Avenue.
Enrichment12-5
r(e, n) �[(e � 1) � (n � 1)]!���
(e � 1)! (n � 1)!
© Glencoe/McGraw-Hill 524 Advanced Mathematical Concepts
The Binomial TheoremTwo ways to expand a binomial are to use either Pascal’striangle or the Binomial Theorem. The Binomial Theoremstates that if n is a positive integer, then the following is true.
(x � y)n � xn � nxn � 1y � �n(
1n
�
�
21)
� xn � 2y2 ��n(n
1�
�
12)(
�
n3� 2)
� xn � 3y3 � . . . � yn
To find individual terms of an expansion, use this form of theBinomial Theorem:
(x � y)n � � �r!(nn�!
r)!� xn � ryr.
Example 1 Use Pascal’s triangle to expand (x � 2y)5.
First, write the series without the coefficients. Theexpression should have 5 � 1, or 6, terms, with the firstterm being x5 and the last term being y5. The exponents of xshould decrease from 5 to 0 while the exponents of y shouldincrease from 0 to 5. The sum of the exponents of each termshould be 5.
x5 � x4y � x3y2 � x2y3 � xy4 � y5 x0 � 1 and y0 � 1
Replace each y with 2y.
x5 � x4(2y) � x3(2y)2 � x2(2y)3 � x(2y)4 � (2y)5
Then, use the numbers in the sixth row of Pascal’s triangleas the coefficients of the terms, and simplify each term.
1 5 10 10 5 1↓ ↓ ↓ ↓ ↓ ↓
(x � 2y)5 � x5 � 5x4(2y) � 10x3(2y)2 � 10x2(2y)3 � 5x(2y)4 � (2y)5
� x5 � 10x4y � 40x3y2 � 80x2y3 � 80xy4 � 32y5
Example 2 Find the fourth term of (5a � 2b)6.
(5a � 2b)6 � � �r!(66�!
r)!� (5a)6 � r(2b)r
To find the fourth term, evaluate the general term for r � 3.Since r increases from 0 to n, r is one less than the numberof the term.
�r!(66�!
r)!�(5a)6 � r(2b)r � �3!(66�!
3)!� (5a)6 � 3(2b)3
� �6 � 53
�!3
4!
� 3!� (5a)3(2b)3
� 20,000a3b3
The fourth term of (5a � 2b)6 is 20,000a3b3.
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
12-6
n
r�0
6
r�0
© Glencoe/McGraw-Hill 525 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
The Binomial Theorem
Use Pascal’s triangle to expand each binomial.
1. (r � 3)5 2. (3a � b)4
Use the Binomial Theorem to expand each binomial.3. (x � 5)4 4. (3x � 2y)4
5. (a � �2�)5 6. (2p � 3q)6
Find the designated term of each binomial expansion.7. 4th term of (2n � 3m)4 8. 5th term of (4a � 2b)8
9. 6th term of (3p � q)9 10. 3rd term of (a � 2�3�)6
11. A varsity volleyball team needs nine members. Of these ninemembers, at least f ive must be seniors. How many of the possible groups of juniors and seniors have at least f ive seniors?
12-6
© Glencoe/McGraw-Hill 526 Advanced Mathematical Concepts
Patterns in Pascal’s TriangleYou have learned that the coefficients in the expansion of (x � y)n
yield a number pyramid called Pascal’s triangle.
Row 1
Row 2
Row 3
Row 4
Row 5
Row 6
Row 7
As many rows can be added to the bottom of the pyramid as you need.This activity explores some of the interesting properties of this famous number pyramid.1. Pick a row of Pascal’s triangle.
a. What is the sum of all the numbers in all the rows above the row you picked?
b. What is the sum of all the numbers in the row you picked?
c. How are your answers for parts a and b related?
d. Repeat parts a through c for at least three more rows of Pascal’s triangle. What generalization seems to be true?
e. See if you can prove your generalization.
2. Pick any row of Pascal’s triangle that comes after the first.a. Starting at the left end of the row, find the sum of the odd
numbered terms.
b. In the same row, find the sum of the even numbered terms.
c. How do the sums in parts a and b compare?
d. Repeat parts a through c for at least three other rows ofPascal’s triangle. What generalization seems to be true?
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
12-6
→
→
→
→
→
→
→
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
© Glencoe/McGraw-Hill 527 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
Special Sequences and SeriesThe value of ex can be approximated by using theexponential series. The trigonometric series can be usedto approximate values of the trigonometric functions. Euler’sformula can be used to write the exponential form of acomplex number and to find a complex number that is thenatural logarithm of a negative number.
Example 1 Use the first five terms of the trigonometricseries to approximate the value of sin ��6� tofour decimal places.
sin x � x � �x33
!� � �x55
!� � �x77
!� � �9x9
!�
Let x � ��6�, or about 0.5236.
sin ��6� � 0.5236 � �(0.5
32!36)3� � �
(0.552!36)5� � �
(0.572!36)7� � �
(0.592!36)9�
sin ��6� � 0.5236 � 0.02392 � 0.00033 � 0.000002 � 0.000000008
sin ��6� � 0.5000 Compare this result to the actual value, 0.5.
Example 2 Write 4 � 4i in exponential form.
Write the polar form of 4 � 4i.Recall that a � bi � r(cos � � i sin �), where r � �a�2��� b�2� and � � Arctan �a
b� when a � 0.
r � �4�2��� (���4�)2� or 4�2�, and a � 4 and b � �4
� � Arctan ��44� or ���4�
4 � 4i � 4�2��cos ����4�� � i sin ����4��� 4�2�e
�i��4�
Thus, the exponential form of 4 � 4i is 4�2�e�i�
�4�.
Example 3 Evaluate ln(�12.4).
ln(�12.4) � ln(�1) � ln(12.4)� i� � 2.5177 Use a calculator to compute ln(12.4).
Thus, ln(�12.4) � i� � 2.5177. The logarithm is a complex number.
12-7
© Glencoe/McGraw-Hill 528 Advanced Mathematical Concepts
Special Sequences and Series
Find each value to four decimal places.
1. ln(�5) 2. ln(�5.7) 3. ln(�1000)
Use the first five terms of the exponential series and a calculatorto approximate each value to the nearest hundredth.
4. e0.5 5. e1.2
6. e2.7 7. e0.9
Use the first five terms of the trigonometric series to approximatethe value of each function to four decimal places. Then, comparethe approximation to the actual value.
8. sin �56�� 9. cos �34
��
Write each complex number in exponential form.
10. 13�cos ��3� � i sin ��3�� 11. 5 � 5i
12. 1 � �3�i 13. �7 � 7�3�i
14. Savings Derika deposited $500 in a savings account with a4.5% interest rate compounded continuously. (Hint: The formulafor continuously compounded interest is A � Pert.)a. Approximate Derika’s savings account balance after 12 years
using the first four terms of the exponential series.
b. How long will it take for Derika’s deposit to double, providedshe does not deposit any additional funds into her account?
PracticeNAME _____________________________ DATE _______________ PERIOD ________
12-7
© Glencoe/McGraw-Hill 529 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
Power SeriesA power series is a series of the form
a0 � a1x � a2x2 � a3x3 � ...
where each ai is a real number. Many functions can be represented by power series. For instance, the function f (x) � ex can be represented by the series
ex � 1 � x � � � ... .
Use a graphing calculator or computer to graph the functions in Exercies 1–4.
1. f2(x) � 1 � x 2. f3(x) � 1 � x �
3. f4(x) � 1 � x � � 4. f5(x) � 1 � x � � �
5. Write a statement that relates the sequence of graphs suggested by Exercies 1–4 and the function y � ex.
6. The series 1 � x2 � x4 � x6 � ... is a power series for which each ai � 1. The series is also a geometric series with first term 1 andcommon ratio x2.a. Find the function that this power series represents.
b. For what values of x does the series give the values of the function in part a?
7. Find a power series representation for the function f (x) � .3
�1 � x2
x4
�4!
x3
�3!
x2
�2!
x3
�3!
x2
�2!
x2
�2!
x3
�3!
x2
�2!
12-7
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
12-8
Sequences and IterationEach output of composing a function with itself is called aniterate. To iterate a function ƒ(x), find the function value ƒ(x0)of the initial value x0. The second iterate is the value of thefunction performed on the output, and so on.
The function ƒ(z) � z2 � c, where c and z are complex numbers,is central to the study of fractal geometry. This type ofgeometry can be used to describe things such as coastlines,clouds, and mountain ranges.
Example 1 Find the first four iterates of the functionƒ(x) � 4x � 1 if the initial value is �1.
x0 � �1x1 � 4(�1) � 1 or �3x2 � 4(�3) � 1 or �11x3 � 4(�11) � 1 or �43x4 � 4(�43) � 1 or �171
The first four iterates are �3, �11, �43, and �171.
Example 2 Find the first three iterates of the functionƒ(z) � 3z � i if the initial value is 1 � 2i.
z0 � 1 � 2iz1 � 3(1 � 2i) � i or 3 � 5iz2 � 3(3 � 5i) � i or 9 � 14iz3 � 3(9 � 14i) � i or 27 � 41i
The first three iterates are 3 � 5i, 9 � 14i, and 27 � 41i.
Example 3 Find the first three iterates of the functionƒ(z) � z2 � c, where c � 2 � i and z0 � 1 � i.
z1 � (1 � i)2 � 2 � i.� 1 � i � i � i2 � 2 � i� 1 � i � i � (�1) � 2 � i i2 � �1� 2 � i
z2 � (2 � i)2 � 2 � i� 4 � 2i � 2i � i2 � 2 � i� 4 � 2i � 2i � (�1) � 2 � i� 5 � 3i
z3 � (5 � 3i)2 � 2 � i� 25 � 15i � 15i � 9i2 � 2 � i� 25 � 15i � 15i � 9(�1) � 2 � i� 18 � 29i
The first three iterates are 2 � i, 5 � 3i, and 18 � 29i.
© Glencoe/McGraw-Hill 530 Advanced Mathematical Concepts
© Glencoe/McGraw-Hill 531 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
Sequences and Iteration
Find the first four iterates of each function using the given initial value. If necessary, round your answers to the nearest hundredth.
1. ƒ(x) � x2 � 4; x0 � 1 2. ƒ(x) � 3x � 5; x0 � �1
3. ƒ(x) � x2 � 2; x0 � �2 4. ƒ(x) � x(2.5 � x); x0 � 3
Find the first three iterates of the function ƒ(z) � 2z � (3 � i) for each initial value.
5. z0 � i 6. z0 � 3 � i
7. z0 � 0.5 � i 8. z0 � �2 � 5i
Find the first three iterates of the function ƒ(z) � z2 � c for eachgiven value of c and each initial value.
9. c � 1 � 2i; z0 � 0 10. c � i; z0 � i
11. c � 1 � i; z0 � �1 12. c � 2 � 3i; z0 � 1 � i
13. Banking Mai deposited $1000 in a savings account. Theannual yield on the account is 5.2%. Find the balance of Mai’saccount after each of the f irst 3 years.
12-8
© Glencoe/McGraw-Hill 532 Advanced Mathematical Concepts
DepreciationTo run a business, a company purchases assets such as equipment orbuildings. For tax purposes, the company distributes the cost of these assets as a business expense over the course of a number ofyears. Since assets depreciate (lose some of their market value) as they get older, companies must be able to figure the depreciationexpense they are allowed to take when they file their income taxes.
Depreciation expense is a function of these three values:1. asset cost, or the amount the company paid for the asset;2. estimated useful life, or the number of years the company can
expect to use the asset;3. residual or trade-in value, or the expected cash value of the
asset at the end of its useful life.
In any given year, the book value of an asset is equal to the asset cost minus the accumulated depreciation. This value represents theunused amount of asset cost that the company may depreciate infuture years. The useful life of the asset is over once its book valueis equal to its residual value.
There are several methods of determining the amount of depreciation in a given year. In the declining-balance method, thedepreciation expense allowed each year is equal to the book value ofthe asset at the beginning of the year times the depreciation rate.Since the depreciation expense for any year is dependent upon thedepreciation expense for the previous year, the process of determining the depreciation expense for a year is an iteration.
The table below shows the first two iterates of the depreciation schedule for a $2500 computer with a residual value of $500 if the depreciation rate is 40%.
1. Find the next two iterates for the depreciation expense function.
2. Find the next two iterates for the end-of-year book value function.
3. Explain the depreciation expense for year 5.
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
12-8
End of Asset Depreciation Book Value atYear Cost Expense End of Year
1 $2500 $1000 $1500(40% of $2500) ($2500 - $1000)
2 $2500 $600 $900(40% of $1500) ($1500 - $600)
© Glencoe/McGraw-Hill 533 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
Mathematical InductionA method of proof called mathematical induction can beused to prove certain conjectures and formulas. The followingexample demonstrates the steps used in proving a summationformula by mathematical induction.
Example Prove that the sum of the first n positiveeven integers is n(n � 1).
Here Sn is defined as 2 � 4 � 6 � . . . � 2n � n(n � 1).
1. First, verify that Sn is valid for the first possible case, n � 1.Since the first positive even integer is 2 and 1(1 � 1) � 2, the formula is valid for n � 1.
2. Then, assume that Sn is valid for n � k.
Sk ⇒ 2 � 4 � 6 � . . . � 2k � k(k � 1). Replace n with k.
Next, prove that Sn is also valid for n � k � 1.
Sk � 1 ⇒ 2 � 4 � 6 � . . . � 2k � 2(k � 1)
� k(k � 1) � 2(k � 1) Add 2(k � 1) to both sides.
We can simplify the right side by adding k(k � 1) � 2(k � 1).
Sk � 1 ⇒ 2 � 4 � 6 � . . . � 2k � 2(k � 1)
� (k � 1)(k � 2) (k � 1) is a common factor.
If k � 1 is substituted into the original formula (n(n � 1)),the same result is obtained.
(k � 1)[(k � 1) � 1] or (k � 1)(k � 2)
Thus, if the formula is valid for n � k, it is also validfor n � k � 1. Since Sn is valid for n � 1, it is alsovalid for n � 2, n � 3, and so on. That is, the formulafor the sum of the first n positive even integers holds.
12-9
© Glencoe/McGraw-Hill 534 Advanced Mathematical Concepts
Mathematical InductionUse mathematical induction to prove that each proposition is validfor all positive integral values of n.
1. �13� � �23� � �33� � . . . � �n3� � �
n(n6� 1)�
2. 5n � 3 is divisible by 4.
PracticeNAME _____________________________ DATE _______________ PERIOD ________
12-9
© Glencoe/McGraw-Hill 535 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
Conjectures and Mathematical InductionFrequently, the pattern in a set of numbers is not immediately evident. Once you make a conjecture about a pattern, you can usemathematical induction to prove your conjecture.
1. a. Graph f (x) � x2 and g(x) � 2x on the axes shown at the right.
b. Write a conjecture that compares n2 and 2n, where n is a positive integer.
c. Use mathematical induction to prove your response from part b.
2. Refer to the diagrams at the right.a. How many dots would there be in the fourth
diagram S4 in the sequence?
b. Describe a method that you can use to determine the number of dots in the fifth diagram S5 based on the number of dots in the fourth diagram, S4. Verify your answer by constructing the fifth diagram.
c. Find a formula that can be used to compute the number of dots in the nth diagram of this sequence.Use mathematical induction to prove your formula iscorrect.
12-9
S1 S2 S3
BLANK
© Glencoe/McGraw-Hill 537 Advanced Mathematical Concepts
Chapter 12 Test, Form 1A
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12Write the letter for the correct answer in the blank at the right of each problem.
1. Find the 15th term in the arithmetic sequence 14, 10.5, 7, . . . . 1. ________A. �21 B. �63 C. 63 D. �35
2. Find the sum of the first 36 terms in the arithmetic series 2. ________�0.2 � 0.3 � 0.8 � . . . .A. 318.6 B. 332.2 C. 307.8 D. 315
3. Find the sixth term in the geometric sequence �3� y3, �3y5, 3�3� y7, . . . . 3. ________A. �27y13 B. 9�3� y13 C. 27y13 D. �9�3� y13
4. Find the sum of the first f ive terms in the geometric series 4. ________��32� � 1 � �23� � . . . .
A. �5554� B. ��55
54� C. �52
57� D. ��52
57�
5. Find three geometric means between ��2� and �4�2�. 5. ________A. 2, �2�2�, 4 B. �2, 2�2�, �4C. 2, 2�2�, 4 D. A or C
6. Find limn→∞ �1 � �
(�n1)n
� . 6. ________
A. 1 B. 0 C. �1 D. does not exist
7. Find the sum of �2�7� � �9� � �3� � . . . . 7. ________
A. �12�(9 � 9 �3�) B. 9 � 9�3� C. �21�(9 � 9 �3�) D. does not exist
8. Write 3.1�2�3� as a fraction. 8. ________
A. �13034303� B. �13
03430� C. �10
3430� D. �1
3034303�
9. Which of the following series is convergent? 9. ________A. �3� � 3 � 3�3� � . . . B. 6�2� � 12 � 12�2� � . . .C. 6�2� � 6 � 3�2� � . . . D. 6�2� � 12 � 12�2� � . . .
10. Which of the following series is divergent? 10. ________
A. 1 � 3��14�� � 9��14��2
� 27��14��3
� . . . B. 1 � 3��15�� � 9��15��2
� 27��15��3
� . . .
C. 1 � 3��17�� � 9��17��2
� 27��17��3
� . . . D. 1 � 3��12�� � 9��12��2
� 27��12��3
� . . .
11. Write �3
k�0���2
1��k
in expanded form and then find the sum. 11. ________
A. ��12� � �14� � �18�; ��78� B. 1 � �12� � �14� � �18�; �18�
C. ��12� � �14� � �18�; �83� D. 1 � �12� � �14� � �18�; �58�
© Glencoe/McGraw-Hill 538 Advanced Mathematical Concepts
12. Express the series �27 � 9 � 3 � 1 � . . . using sigma notation. 12. ________
A. �∞
k�0�3k B. �
3
k�0�27���13��
k
C. �∞
k�0�27���13��
kD. �
∞
k�027���13��
k
13. The expression 81p4 � 108p3r3 � 54p2r6 � 12pr9 � r12 is the 13. ________expansion of which binomial?A. ( p � 3r3)4 B. (3p � r3)4 C. (3p3 � r)4 D. (3p � 3r3)4
14. Find the fifth term in the expansion of (3x2 � �y�)6. 14. ________A. 135x4y2 B. 45x4y2 C. �135x4y2 D. �45x4y2
15. Use the first f ive terms of the trigonometric series to find the value 15. ________of sin �1
�2� to four decimal places.
A. 0.2618 B. 0.2588 C. 0.7071 D. 0.2648
16. Find ln (�91.48). 16. ________A. 4.5161 B. i� � 4.5161 C. i� � 4.5161 D. �4.5161
17. Write 3 � �3�i in exponential form. 17. ________A. 9ei�
116�� B. 9ei�
53�� C. 2�3�ei�
116�� D. 2�3�ei�
53��
18. Find the first three iterates of the function ƒ(z) � �z � i for 18. ________z0 � 2 � 3i.A. 2 � 2i, 2 � 3i, 2 � 2i B. �2 � 2i, 2 � 3i, �2 � 2iC. 2 � 3i, �2 � 2i, 2 � 3i D. �2 � 2i, 2 � 3i, �2 � 2i
19. Find the first three iterates of the function ƒ(z) � z2 � c for 19. ________c � 1 � 2i and z0 � 1 � i.A. �1 � 4i, �15 � 8i, 219 � 194iB. �1 � 4i, �16 � 6i, 220 � 192iC. �1 � 4i, �16 � 6i, 221 � 194iD. �1 � 4i, �16 � 6i, 219 � 194i
20. Suppose in a proof of the summation formula 20. ________1 � 5 � 25 � . . . � 5n�1 � �14�(5n � 1) by mathematical induction,you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k � 1) � �14� (5k�1 � 1)
B. 1 � 5 � 25 � . . . � 5k � 5k�1 � �14� (5k � 1) � 5k�1�1
C. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k�1 � 1) � 5k�1�1
D. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k � 1) � 5k�1�1
Bonus Solve �6
n�0(3n � 2x) � 7 for x. Bonus: ________
A. �121� B. 8 C. 4 D. �13
9�
Chapter 12 Test, Form 1A (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12
© Glencoe/McGraw-Hill 539 Advanced Mathematical Concepts
Chapter 12 Test, Form 1B
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right ofeach problem.
1. Find the 27th term in the arithmetic sequence �8, 1, 10, . . . . 1. ________A. 174 B. 242 C. 235 D. 226
2. Find the sum of the first 20 terms in the arithmetic series 2. ________14 � 3 � 8 � . . . .A. �195 B. �1810 C. 195 D. 1810
3. Find the sixth term in the geometric sequence 11, �44, 176, . . . . 3. ________A. 11,264 B. �11,264 C. 45,056 D. �45,056
4. Find the sum of the first five terms in the geometric series 4. ________2 � �43� � �89� � . . . .
A. �8515� B. �12
37� C. �18
110� D. �28
715�
5. Find three geometric means between ��23� and �54. 5. ________A. 2, 6, 18 B. �2, 6, �18 C. 2, �6, 18 D. A or C
6. Find lim �5n43n�
3 �7n
72n�
2
3�. 6. ________
A. �54� B. 0 C. �45� D. does not exist
7. Find the sum of �151� � �35
35� � �6
9095� � . . . . 7. ________
A. �17201� B. ��17
201� C. �27
2� D. does not exist
8. Write 0.1�2�3� as a fraction. 8. ________A. �43
13� B. �3
43133� C. �3
4313� D. �34
313�
9. Which of the following series is convergent? 9. ________A. 7.5 � 1.5 � 0.3 � . . . B. 1.2 � 3.6 � 10.8 � . . .C. 1.2 � 3.6 � 10.8 � . . . D. �2.5 � 2.5 � 2.5 � . . .
10. Which of the following series is divergent? 10. ________A. �3
12� � �6
12� � �9
12� � . . . B. �23
2� � �26
4� � �29
6� � . . .
C. �13�12� � �23
�23� � �33
�34� � . . . D. �0.
352� � �0.
654� � �0.
956� � . . .
11. Write �5��23��kin expanded form and then find the sum. 11. ________
A. 5��23��2� ��23��2
� ��23��2; �29
8� B. ��5 3� 2��2
� ��5 3� 2��3
� ��5 3� 2��4
; �15
8,7100
�
C. 5��23��1� 5��23��2
� 5��23��3; �12
970� D. 5��23��2
� 5��23��3� 5��23��4
; �38810�
Chapter
12
k�2
4
© Glencoe/McGraw-Hill 540 Advanced Mathematical Concepts
12. Express the series 0.7 � 0.007 � 0.00007 � . . . using sigma notation. 12. ________
A. �0.7(10)k � 1 B. �7(10)1 � 2k C. �7(10)1 � k D. �0.7(10)�k
13. The expression 243c5 � 810c4d � 1080c3d2 � 720c2d3 � 240cd4 � 32d5 13. ________is the expansion of which binomial?A. (3c � d)5 B. (c � 2d)5 C. (2c � 3d)5 D. (3c � 2d)5
14. Find the third term in the expansion of (3x � y)6. 14. ________A. 1215x4y2 B. 1215x2y4 C. �1215x2y4 D. �1215x4y2
15. Use the first five terms of the exponential series 15. ________ex � 1 � x � �x2
2
!� ��x33
!� � �x44
!� � . . . to approximate e3.9.
A. 39.40 B. 24.01 C. 32.03 D. 90.11
16. Find ln (�102). 16. ________A. 4.6250 B. i� � 4.6250 C. i� � 4.6250 D. �4.6250
17. Write 15�3� � 15i in exponential form. 17. ________
A. 30ei�11
6�� B. 30ei�
56�� C. 30ei�
76�� D. 15ei�
116��
18. Find the first three iterates of the function ƒ(z) � �2z for z0 � 1 � 3i. 18. ________A. 2 � 6i, 4 � 12i, 8 � 24i B. �2 � 6i, 4 � 12i, �8 � 24iC. �2 � 6i, 4 � 12i, 8 � 24i D. 2 � 6i, �4 � 12i, 8 � 24i
19. Find the first three iterates of the function ƒ(z) � z2 � c for c � i 19. ________and z0 � 1.A. 1 � i, 2 � 3i, �5 � 13i B. 1 � i, �3i, �9 � iC. 1 � i, �3i, 9 � i D. 1 � i, �2i, �4 � i
20. Suppose in a proof of the summation formula 7 � 9 � 11 � . . . � 20. ________2n � 5 � n(n � 6) by mathematical induction, you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 7 � 9 � 11 � . . . � 2k � 5 � 2(k � 1) � 5 � k(k � 6) � (k � 1)(k � 1 � 6)B. 7 � 9 � 11 � . . . � 2(k � 1) � 5 � k(k � 6)C. 7 � 9 � 11 � . . . � 2k � 5 � k(k � 6)D. 7 � 9 � 11 � . . . � 2k � 5 � 2(k � 1) � 5 � k(k � 6) � 2(k � 1) � 5
Bonus If a1, a2, a3, . . ., an is an arithmetic sequence, where an 0, Bonus: ________then �a
11�, �a
12�, �a
13�, . . ., �a
1n� is a harmonic sequence. Find one
harmonic mean between �12� and �18�.
A. �14� B. �15� C. �16� D. �156�
Chapter 12 Test, Form 1B (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12
∞
k�1
∞
k�1
∞
k�1
∞
k�1
© Glencoe/McGraw-Hill 541 Advanced Mathematical Concepts
Chapter 12 Test, Form 1C
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right ofeach problem.
1. Find the 21st term in the arithmetic sequence 9, 3, �3, . . . . 1. ________A. �111 B. �129 C. �117 D. �126
2. Find the sum of the first 20 terms in the arithmetic series 2. ________�6 � 12 � 18 � . . . .A. �2520 B. �1266 C. �1140 D. �1260
3. Find the 10th term in the geometric sequence �2, 6, �18, . . . . 3. ________A. 118,098 B. �118,098 C. 39,366 D. �39,366
4. Find the sum of the first eight terms in the geometric series 4. ________�4 � 8 � 16 � . . . .A. �342 B. �1020 C. �340 D. 340
5. Find one geometric mean between 2 and 32. 5. ________A. �16 B. 8 C. 12 D. 4
6. Find limn→∞
�n3
n�2
5� . 6. ________
A. �23� B. �5 C. �85� D. does not exist
7. Find the sum of 16 � 4 � 1 � . . . . 7. ________
A. 64 B. �654� C. 20 D. does not exist
8. Write 0.8� as a fraction. 8. ________
A. �98989� B. �9
8� C. �989� D. �8
9�
9. Which of the following series is convergent? 9. ________A. 8 � 8.8 � 9.68 � . . . B. 8 � 6 � 4 � . . .C. 8 � 2.4 � 0.72 � . . . D. 8 � 8 � 8 � . . .
10. Which of the following series is divergent? 10. ________
A. 1 � �21
2� � �
21
4� � �
21
6� � . . . B. 1 � �
31
2� � �
31
4� � �
31
6� � . . .
C. 1 � �21
2� � �
21
4� � �
21
6� � . . . D. 1 � ��32��
2� ��32��
4� ��32��
6� . . .
11. Write �4
k�13k�1 in expanded form and then find the sum. 11. ________
A. 1 � 3 � 9 � 27; 40 B. 1 � �13� � �19� � �217�; �42
07�
C. 3 � 9 � 27 � 81; 120 D. 0 � 2 � 8 � 26; 36
Chapter
12
© Glencoe/McGraw-Hill 542 Advanced Mathematical Concepts
12. Express the series 5 � 9 � 13 � . . . � 101 using sigma notation. 12. ________
A. �∞
k�1(4k � 1) B. �
25
k�1(4k � 1) C. �
25
k�1(4k � 1) D. �
24
k�1(4k � 1)
13. The expression 32x5 � 80x4 � 80x3 � 40x2 � 10x � 1 is the 13. ________expansion of which binomial?A. (2x � 1)5 B. (x � 2)5 C. (2x � 2)5 D. (2x � 1)5
14. Find the fourth term in the expansion of (3x � y)7. 14. ________A. 105x4y3 B. 420x4y3 C. 1701x4y3 D. 2835x4y3
15. Use the first f ive terms of the exponential series 15. ________ex � 1 � x � �x2
2
!� � �x33
!� � �x44
!� � . . . to approximate e5.
A. 65.375 B. 148.41 C. 48.41 D. 76.25
16. Find ln (�21). 16. ________A. 3.0445 B. i� � 3.0445 C. i� � 3.0445 D. �3.0445
17. Write 1 � i in exponential form. 17. ________A. �2� e i�
�4� B. �2�ei�
74�� C. ei�
74�� D. ei�
�4�
18. Find the first three iterates of the function ƒ(z) � z � i for z0 � 1. 18. ________A. 1, 1 � i, 1 � 2i B. 1 � i, 2 � 2i, 3 � 3iC. 1 � i, 1 � 2i, 1 � 3i D. 1 � i, 1 � i, 1 � i
19. Find the first three iterates of the function ƒ(z) � z2 � c for c � i 19. ________and z0 � i.A. �1 � i, i, �1�i B. �1 � i, 3i, �9C. 1 � i, �3i, 9 � i D. �1 � i, 2i, �4 � i
20. Suppose in a proof of the summation formula 20. ________1 � 5 � 9 � . . . � 4n � 3 � n(2n � 1) by mathematical induction,you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 1 � 5 � 9 � . . . � 4k � 3 � 4(k � 1) � 3 � k(2k � 1) � 4(k � 1) � 3B. 1 � 5 � 9 � . . . � 4k � 3 � k(2k � 1) � 4(k � 1) � 3C. 1 � 5 � 9 � . . . � 4k � 3 � k(2k � 1)D. 1 � 5 � 9 � . . . � 4k � 3 � 4(k � 1) � 3 � k(2k � 1) � (k � 1)[2(k � 1) � 1]
Bonus If a1, a2, a3, . . . , an is an arithmetic sequence, where an 0, Bonus: ________
then �a1
1�, �a
12�, �a
13�, . . . , �a
1n� is a harmonic sequence. Find one
harmonic mean between 2 and 3.
A. �25� B. �52� C. �152� D. �15
2�
Chapter 12 Test, Form 1C (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12
© Glencoe/McGraw-Hill 543 Advanced Mathematical Concepts
Chapter 12 Test, Form 2A
NAME _____________________________ DATE _______________ PERIOD ________
1. Find d for the arithmetic sequence in which a1 � 14 and 1. __________________a28 � 32.
2. Find the 15th term in the arithmetic sequence 2. __________________11�45�, 10 �25�, 9, 7�35�, . . . .
3. Find the sum of the first 27 terms in the arithmetic series 3. __________________35.5 � 34.3 � 33.1 � 31.9 � . . . .
4. Find the ninth term in the geometric sequence 25, 10, 4, . . . . 4. __________________
5. Find the sum of the first eight terms in the geometric series 5. __________________�15� � 2 � 20 � . . . .
6. Form a sequence that has three geometric means between 6. __________________6 and 54.
7. Find limn→∞
�9n
133n
�
4
5�
n25n
�
2
4� or state that the limit does not exist. 7. __________________
8. Find the sum of the series 6�2� � 6 � 3�2� � 3 � . . . or 8. __________________state that the sum does not exist.
9. Write 0.06�4� as a fraction. 9. __________________
Determine whether each series is convergent or divergent.
10. �2�2�
1� 13� � �
2�2�1
� 23� � �2�2�
1� 33� � . . . 10. __________________
11. �211� � �22
2� � �23
3� � . . . 11. __________________
Chapter
12
© Glencoe/McGraw-Hill 544 Advanced Mathematical Concepts
12. Write �7
k�227���13��
k�2in expanded form and then find 12. __________________
the sum.
13. Express the series �31�09� � �31
�211� � �31
�413� � . . . � �32
�423� 13. __________________
using sigma notation.
14. Use the Binomial Theorem to expand (1 � �3�)5. 14. __________________
15. Find the fifth term in the expansion of (3x3 � 2y2)5. 15. __________________
16. Use the first f ive terms of the exponential series 16. __________________to approximate e2.7.
17. Find ln (�12.7) to four decimal places. 17. __________________
18. Find the first three iterates of the function ƒ(z) � 3z � 1 18. __________________for z0 � 2 � i.
19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � 1 � i and z0 � 2i.
20. Use mathematical induction to prove that 20. __________________1 � 5 � 25 � . . . � 5n�1 � �4
1� (5n � 1). Write your proof on a separate piece of paper.
Bonus If ƒ(z) � z2 � z � c is iterated with an initial Bonus: __________________value of 3 � 4i and z1 � 4 � 11i, f ind c.
Chapter 12 Test, Form 2A (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12
© Glencoe/McGraw-Hill 545 Advanced Mathematical Concepts
Chapter 12 Test, Form 2B
NAME _____________________________ DATE _______________ PERIOD ________
1. Find d for the arithmetic sequence in which a1 � 6 1. __________________and a13 � �42.
2. Find the 40th term in the arithmetic sequence 2. __________________7, 4.4, 1.8, �0.8, . . . .
3. Find the sum of the first 30 terms in the arithmetic series 3. __________________10 � 6 � 2 � 2 � . . . .
4. Find the ninth term in the geometric sequence �217�, �19�, �13�, . . . . 4. __________________
5. Find the sum of the first eight terms in the geometric series 5. __________________64 � 32 � 16 � 8 � . . . .
6. Form a sequence that has three geometric means between 6. __________________�4 and �324.
7. Find lim �2nn�3
1� or state that the limit does not exist. 7. __________________
8. Find the sum of the series 12 � 8 � �136� � . . . or state 8. __________________
that the sum does not exist.
9. Write 8.1�8� as a fraction. 9. __________________
Determine whether each series is convergent or divergent.
10. �41� � �17� 2� � �1 �
120
� 3� � . . . 10. __________________
11. �311� � �32
2� � �33
3� � . . . 11. __________________
Chapter
12
© Glencoe/McGraw-Hill 546 Advanced Mathematical Concepts
12. Write �(k � 1)(k � 2) in expanded form and then find the 12. __________________sum.
13. Express the series �1 2� 0� � �2 3
� 1� � �3 4� 2� � . . . � �10
11� 9� using 13. __________________
sigma notation.
14. Use the Binomial Theorem to expand (2p � 3q)4. 14. __________________
15. Find the fifth term in the expansion of (4x � 2y)7. 15. __________________
16. Use the first five terms of the cosine series 16. __________________cos x � 1 � �x2
2
!� � �x44
!� � �x66
!� � �x88
!� � . . . to approximate
the value of cos �4�� to four decimal places.
17. Find ln (�13.4) to four decimal places. 17. __________________
18. Find the first three iterates of the function ƒ(z) � 0.5z 18. __________________for z0 � 4 � 2i.
19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � 2i and z0 � 1.
20. Use mathematical induction to prove that 7 � 9 � 11 � . . . � 20. __________________(2n � 5) � n(n � 6). Write your proof on a separate piece of paper.
Bonus Find the sum of the coefficients of the expansion Bonus: __________________of (x � y)7.
Chapter 12 Test, Form 2B (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
123
k�0
© Glencoe/McGraw-Hill 547 Advanced Mathematical Concepts
Chapter 12 Test, Form 2C
NAME _____________________________ DATE _______________ PERIOD ________
1. Find d for the arithmetic sequence in which a1 � 5 and 1. __________________a12 � 38.
2. Find the 31st term in the arithmetic sequence 2. __________________9.3, 9, 8.7, 8.4, . . . .
3. Find the sum of the first 23 terms in the arithmetic 3. __________________series 6 � 11 � 16 � 21 � . . . .
4. Find the fifth term in the geometric sequence 4. __________________�10, �40, �160, . . . .
5. Find the sum of the first 10 terms in the geometric 5. __________________series 3 � 6 � 12 � 24 � . . . .
6. Form a sequence that has two geometric means 6. __________________between 9 and �13�.
7. Find limn→∞
�n2
2n�
2
1� or state that the limit does not exist. 7. __________________
8. Find the sum of the series �112� � �12� � 3 � . . . or state 8. __________________
that the sum does not exist.
9. Write 0.5�3� as a fraction. 9. __________________
Determine whether each series is convergent or divergent.
10. �21
0� � �
21
2� � �
21
4� � . . . 10. __________________
11. �211� � �22
2� � �23
3� � . . . 11. __________________
Chapter
12
© Glencoe/McGraw-Hill 548 Advanced Mathematical Concepts
12. Write �7
k�43k in expanded form and then find the sum. 12. __________________
13. Express the series �12� 2� � �2 4
� 3� � �36� 4� � . . . � �81
�69� using 13. __________________
sigma notation.
14. Use the Binomial Theorem to expand (2p � 1)4. 14. __________________
15. Find the fourth term in the expansion of (2x � 3y)4. 15. __________________
16. Use the first f ive terms of the sine series sin x � 16. __________________x � �x3
3
!� � �x55
!� � �x77
!� � �x99
!� � . . . to f ind the value of
sin ��5� to four decimal places.
17. Find ln (�58) to four decimal places. 17. __________________
18. Find the first three iterates of the function ƒ(z) � 2z for 18. __________________z0 � 1 � 4i.
19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � i and z0 � 1.
20. Use mathematical induction to prove that 1 � 5 � 9 � . . . � 4n � 3 � n(2n � 1). Write your proof 20. __________________on a separate piece of paper.
Bonus Find the sum of the coefficients of the expansion Bonus: __________________of (x � y)5.
Chapter 12 Test, Form 2C (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12
© Glencoe/McGraw-Hill 549 Advanced Mathematical Concepts
Chapter 12 Open-Ended Assessment
NAME _____________________________ DATE _______________ PERIOD ________
Instructions: Demonstrate your knowledge by giving a clear, concisesolution to each problem. Be sure to include all relevant drawingsand justify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.
1. a. Write a word problem that involves an arithmetic sequence.Write the sequence and solve the problem. Tell what the answerrepresents.
b. Find the common difference and write the nth term of thearithmetic sequence in part a.
c. Find the sum of the first 12 terms of the arithmetic sequence inpart a. Explain in your own words why the formula for the sum ofthe first n terms of an arithmetic series works.
d. Does the related arithmetic series converge? Why or why not?
2. a. Write a word problem that involves a geometric sequence. Writethe sequence and solve the problem. Tell what the answerrepresents.
b. Find the common ratio and write the nth term of the geometricsequence in part a.
c. Find the sum of the first 11 terms of the sequence in part a.
d. Describe in your own words a test to determine whether a geometric series converges. Does the geometric series in part a converge?
3. a. Explain in your own words how to use mathematical induction toprove that a statement is true for all positive integers.
b. Use mathematical induction to prove that the sum of the first n terms of a geometric series is given by the formula
Sn � �a1
1�
�
ar1r
n
�, where r 1.
4. Find the fourth term in the expansion of ���
y2
x�� � �
�yx�� �
6.
Chapter
12
© Glencoe/McGraw-Hill 550 Advanced Mathematical Concepts
1. Find the 20th term in the arithmetic sequence 1. __________________15, 21, 27, . . . .
2. Find the sum of the first 25 terms in the arithmetic 2. __________________series 11 � 14 � 17 � 20 � . . . .
3. Find the 12th term in the geometric sequence 3. __________________2�4, 2�3, 2�2, . . . .
4. Find the sum of the first 10 terms in the geometric 4. __________________series 2 � 6 � 18 � 54 � . . . .
5. Write a sequence that has two geometric means 5. __________________between 64 and �8.
6. Find limn→∞
�2n
n2
2
�
�
31n
� or state that the limit 6. __________________
does not exist.
7. Find the sum of the series �18� � �14� � �12� � . . . or state 7. __________________that the sum does not exist.
8. Write 0.6�3� as a fraction. 8. __________________
Determine whether each series is convergent or divergent.
9. 5 � �15�
2
2� � �1 �52
3
� 3� � �1 � 25�
4
3 � 4� � . . . 9. __________________
10. �222� � �23
3� � �24
4� � �25
5� � . . . 10. __________________
Chapter 12 Mid-Chapter Test (Lessons 12-1 through 12-4)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12
1. Find the 11th term in the arithmetic sequence 1. __________________�3� � �5�, 0, ��3� � �5�, . . . .
2. Find n for the sequence for which an � 19, a1 � �13, 2. __________________and d � 2.
3. Find the sum of the first 17 terms in the arithmetic 3. __________________series 4.5 � 4.7 � 4.9 � . . . .
4. Find the fifth term in the geometric sequence for which 4. __________________a3 � �5� and r � 3.
5. Find the sum of the first six terms in the geometric 5. __________________series 1 � 1.5 � 2.25 � . . . .
6. Write a sequence that has one geometric mean 6. __________________between �13� and �2
57�.
Chapter 12, Quiz B (Lessons 12-3 and 12-4)
NAME _____________________________ DATE _______________ PERIOD ________
Chapter 12, Quiz A (Lessons 12-1 and 12-2)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 551 Advanced Mathematical Concepts
Chapter
12
Chapter
12Find each limit, or state that the limit does not exist. 1. __________________
1. limn→∞
�2n
32n�
4
5� 2. lim
n→∞�(2n �
21n)(
2
n � 2)� 3. lim
n→∞�nn
2�
�
14
�2. __________________
3. __________________
Find the sum of each series, or state that the sum 4. __________________does not exist.
4. �12� � �14� � �18� � �116� � . . . 5. ��35� � 1 � �53� � . . . 5. __________________
6. Write the repeating decimal 0.4�5� as a fraction. 6. __________________
Determine whether each series is convergent or divergent.7. 0.002 � 0.02 � 0.2 � . . . 7. __________________
8. �58� � �59� � �150� � �1
51� � . . . 8. __________________
9. �11� 2� � �2
1� 3� � �3
1� 4� � �4
1� 5� � . . . 9. __________________
10. �12� 2� � �2
3� 3� � �3
4� 4� � . . . 10. __________________
1. Write �4
n�2�2n�1 � �12�� in expanded form and then find the sum. 1. __________________
2. Express the series �1861� � �2
87� � �49� � . . . using sigma notation. 2. __________________
3. Express the series 1 � 2 � 3 � 4 � 5 � 6 � . . . � 199 � 200 3. __________________using sigma notation.
4. Use the Binomial Theorem to expand (3a � d)4. 4. __________________
5. Use the first five terms of the exponential series 5. __________________ex � 1 � x � �x2
2
!� � �x33
!� � �x44
!� � . . . to approximate e4.1
to the nearest hundredth.
6. Use the first five terms of the trigonometric series 6. __________________sin x � x � �x3
3
!� � �x55
!� � �x77
!� � �x99
!� � . . . to approximate sin �
�3� to four decimal places.
Chapter 12, Quiz D (Lessons 12-8 and 12-9)
NAME _____________________________ DATE _______________ PERIOD ________
Chapter 12, Quiz C (Lessons 12-5 and 12-6)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 552 Advanced Mathematical Concepts
Chapter
12
Chapter
121. Find the first four iterates of the function ƒ(x) � �1
10� x � 1 1. __________________
for x0 � 1.
2. Find the first three iterates of the function ƒ(z) � 2z � i 2. __________________for z0 � 3 � i.
3. Find the first three iterates of the function ƒ(z) � z2 � c 3. __________________for c � �1 � 2i and z0 � i.
4. Use mathematical induction to prove that 4. __________________1 � 3 � 5 � . . . � (2n � 1) � n2. Write your proof on a separate piece of paper.
5. Use mathematical induction to prove that 5. __________________5 � 11 � 17 � . . . � (6n � 1) � n(3n � 2). Write your proof on a separate piece of paper.
© Glencoe/McGraw-Hill 553 Advanced Mathematical Concepts
Chapter 12 SAT and ACT Practice
NAME _____________________________ DATE _______________ PERIOD ________Chapter
12After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.
Multiple Choice1. In a basket of 80 apples, exactly 4 are
rotten. What percent of the apples arenot rotten?A 4%B 5%C 20%D 95%E 96%
2. Which grade had the largest percentincrease in the number of studentsfrom 1999 to 2000?
A 8B 9C 10D 11E 12
3. Find the length of a chord of a circle if the chord is 6 units from the center andthe length of the radius is 10 units.A 4B 8C 16D 2�3�4�E 4�3�4�
4. A chord of length 16 is 4 units fromthe center of a circle. Find the diameter.A 2�5�B 4�5�C 8�5�D 4�3�E 8�3�
5. If x � y � 4 and 2x � y � 5, then x � 2y �A 1B 2C 4D 5E 6
6. If �3kx15
�k
36� � 1 and x � 4, then k �
A 2B 3C 4D 8E 12
7. If 12% of a class of 25 students do nothave pets, how many students in theclass do have pets?A 3B 12C 13D 20E 22
8. In a senior class there are 400 boysand 500 girls. If 60% of the boys and50% of the girls live within 1 mile ofschool, what percent of the seniors donot live within 1 mile of school?A about 45.6%B about 54.4%C about 55.5%D about 44.4%E about 61.1%
9. In �ABC below, if BC � BA, which ofthe following is true?A x � yB y � zC y � xD y � xE z � x
Grade 8 9 10 11 121999 60 55 65 62 602000 80 62 72 72 70
© Glencoe/McGraw-Hill 554 Advanced Mathematical Concepts
Chapter 12 SAT and ACT Practice (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
1210. Which is the measure of each angle of
a regular polygon with r sides?A �1r�(360°)B (r � 2)180°C 360°D �1r�(r � 2)180°E 60°
11. A tractor is originally priced at $7000.The price is reduced by 20% and thenraised by 5%. What is the net reduction in price?A $5950 B $5880C $1400 D $1120E $1050
12. The original price of a camera provided a profit of 30% above thedealer’s cost. The dealer sets a newprice of $195, a 25% increase abovethe original price. What is the dealer’s cost?A $243.75 B $202.80C $156.00 D $120.00E None of these
13. Segments of the lines x � 4, x � 9,y � �5, and y � 4 form a rectangle.What is the area of this rectangle insquare units?A 6 B 10C 18 D 20E 45
14. In the figure below, the coordinates ofA are (4, 0) and of C are (15, 0). Findthe area of �ABC if the equation ofAB��� is 2x � y � 8.A 100 units2
B 121 units2
C 132 units2
D 144 units2
E 169 units2
15. In 1998, Bob earned $2800. In 1999,his earnings increased by 15%. In2000, his earnings decreased by 15%from his earnings in 1999. What werehis earnings in 2000?A $2800.00 B $2380.98C $2381.15 D $2737.00E None of these
16. Londa is paid a 15% commission onall sales, plus $8.50 per hour. Oneweek, her sales were $6821.29. Howmany hours did she work to earn$1371.69?A 68.3 B 41.0C 120.4 D 28.2E None of these
17–18. Quantitative ComparisonA if the quantity in Column A is
greaterB if the quantity in Column B is
greaterC if the two quantities are equalD if the relationship cannot be
determined from the informationgiven
Column A Column B
17. x > 0
18. One day, 90% of the girls and 80% ofthe boys were present in class.
19. Grid-In How many dollars must beinvested at a simple-interest rate of7.2% to earn $1440 in interest in 5 years?
20. Grid-In Seventy-two is 150% ofwhat number?
0.75xx plus an increase
of 75% of x
Number ofboys absent
Number ofgirls absent
© Glencoe/McGraw-Hill 555 Advanced Mathematical Concepts
Chapter 12 Cumulative Review (Chapters 1-12)
NAME _____________________________ DATE _______________ PERIOD ________
1. Solve the system by using a matrix equation. 1. __________________2x � 3y � 11y � 12 � x
2. Determine whether ƒ(x) � �xx2 �
�35x� is continuous at x � 3. 2. __________________
Justify your response using the continuity test.
3. Determine the binomial factors of 2x3 � x2 � 13x � 6. 3. __________________
4. Write an equation of the sine function with amplitude 1, 4. __________________period �23
��, phase shift �1�5�, and vertical shift 2.
5. Find the distance between the parallel lines 2x � 5y � 10 5. __________________and 2x � 5y � �5.
6. A 300-newton force and a 500-newton force act on the 6. __________________same object. The angle between the forces measures 95°.Find the magnitude and direction of the resultant force.
7. Find the product 4�cos �23�� � i sin �23
��� � 3�cos �56�� � i sin �56
���. 7. __________________Then write the result in rectangular form.
8. Write the equation of the ellipse 6x2 � 9y2 � 54 after a 8. __________________rotation of 45° about the origin.
9. If $1500 is invested in an account bearing 8.5% interest 9. __________________compounded continuously, find the balance of the accountafter 18 months.
10. Express the series �33� � �65� � �97� � �192� � . . . � �32
01� using 10. __________________
sigma notation.
Chapter
12
BLANK
© Glencoe/McGraw-Hill 557 Advanced Mathematical Concepts
Trigonometry Semester Test
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right ofeach problem.
1. Solve 2x�1 � 17.6. Round your answer to the nearest hundredth. 1. ________A. 1.99 B. 3.14 C. �2.45 D. �3.84
2. Simplify i29 � i20. 2. ________A. 0 B. �2 C. �1 � i D. �2i
3. Find the third iterate of the function ƒ(x) � 2x � 1, if the initial 3. ________value is x0 � 3.A. 15 B. 31 C. 7 D. 12
4. Find the nineteenth term in the arithmetic sequence 10, 7, 4, 1, .... 4. ________A. 102 B. 0 C. �47 D. �44
5. If tan � � � �14� and � has its terminal side in Quandrant II, find the 5. ________exact value of sin �.A. �4 B. ��1�7� C. 2 D. �
�11�77�
�
6. Which expression is equivalent to tan ��2�
� ���? 6. ________
A. sin � B. cot � C. �cos � D. sec �
7. Find the ordered pair that represents the vector from A(1, �2) 7. ________to B(2, 3).A. 2, 3� B. 1, 2� C. 1, 5� D. 3, 1�
8. Which series is divergent? 8. ________A. 1 � �
21
3� � �
31
3� � �
41
3� � . . . B. 1 � �1
2� � �1
3� � . . .
C. 1 � �12� � �14� � �18� � . . . D. 2 � �23� � �29� � �227� � . . .
9. Which represents the graph of (3, 45°)? 9. ________A. B.
C. D.
© Glencoe/McGraw-Hill 558 Advanced Mathematical Concepts
10. Express x�29� y�
13� using radicals. 10. ________
A. �3
x�2y� B. �9
x�2y� C. �9
x�2y�3� �3
x�3y�
11. Which expression is equivalent to sin2 � for all values of �? 11. ________A. sin ��
�2� � �� B. 1 C. 2 sin � cos � D. 1 � cos2 �
12. Find the inner product of v� and w� if v� � 1, 2, 0� and w� � 3, �2, 1�. 12. ________A. 3 B. 2 C. �1 D. 1
13. Simplify (1 � i)(�2 � 2i). 13. ________A. 3 � 2i B. 2 � i C. �2 � 3i D. �4
14. Which equation is a trigonometric identity? 14. ________A. sin 2� � sin � cos � B. tan � � �s
cions���
C. tan 2� � �1
2�
ttaann
�2 �
� D. cos 2� � 4 cos2 � � 1
15. Write MN� as the sum of unit vectors for M(�11, 6, �7) and 15. ________N(4, �3, �15).A. �7i� � 3j� � 22k�� B. 15i� � 9j� � 8k��
C. �15i� � 9j�� 8k�� D. 7i� � 9j� � 8k��
16. Express �5
x�25�y�2� using rational exponents. 16. ________
A. x5y�52� B. x2y2 C. x5y�5
2� D. x�
12�y�5
2�
17. Express (x2 y3)�23� using radicals. 17. ________
A. �2
x�4y�6� B. �2
x�6y�9� C. �3
x�2y�3� D. �3
x�4y�6�
18. Which polar equation represents a rose? 18. ________A. r � 3� B. r � 3 � 3 sin �C. r � 3 cos 2� D. r2 � 4 cos 2�
19. Use a sum or difference identity to find the exact value of cos 105°. 19. ________
A. ��42�� B. �
�2� �4
�6�� C. �
�2� �4
�6�� D. �
�2� �
2�3�
�
Trigonometry Semester Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 559 Advanced Mathematical Concepts
Trigonometry Semester Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
20. Write the equation of the line 2x � 3y � 6 in parametric form. 20. ________
A. x � t; y � ��23�t � 2 B. x � 2t; y � 2t � 3
C. x � t; y � ��32�t � 2 D. x � 3t; y � t � 2
21. Find the sum of the geometric series �13� � �19� � �217� � . . . . 21. ________
A. �27� B. �12� C. �23� D. 1
22. Which is the graph of the equation � � �34��? 22. ________
A. B.
C. D.
23. Solve log6 x � 2. 23. ________A. 3 B. 36 C. 12 D. 4
24. Express 0.3 � 0.03 � 0.003 � . . . using sigma notation. 24. ________
A. �∞
k�1
3 � 10�k B. �∞
k�1
0.3 � 10�k
C. �∞
k�0
3 � 10�k D. �∞
k�1
3 � 10�2k
25. Andre kicks a soccer ball with an initial velocity of 48 feet per 25. ________second at an angle of 17° with the horizontal. After 0.35 second,what is the height of the ball?A. 16.07 ft B. 4.91 ft C. 14.11 ft D. 2.95 ft
© Glencoe/McGraw-Hill 560 Advanced Mathematical Concepts
26. Find v� � w� if v� � 2, �4, 6� and w� � 2, 1, �1�. 26. __________________
27. Write an equation in slope-intercept form of the line whose 27. __________________parametric equations are x � 4t � 3 and y � �2t � 7.
28. Use a calculator to find antiln (�0.23) to the nearest 28. __________________hundredth.
29. Find the first four terms of the geometric sequence for 29. __________________which a9 � 6561 and r � 3.
30. Find the polar coordinates of the point with rectangular 30. __________________coordinates (2, 2).
31. Find the rectangular coordinates of the point with polar 31. __________________coordinates ��2�, �
34���.
32. Express cos 840° as a trigonometric function of an angle 32. __________________in Quadrant II.
33. Solve 2 sin2 x � sin x � 0 for principle values of x. 33. __________________Express in degrees.
34. Find the distance between the point P(1, 0) and the line 34. __________________with equation 2x � 3y � �2.
35. If cos � � �34� and � has its terminal side in Quadrant I, find 35. __________________the exact value of sin �.
36. Express 2 � 4 � 6 � 8 � 10 using sigma notation. 36. __________________
37. Find limn→∞
�(3n � 2
n)2(n � 5)�. 37. __________________
38. Use a calculator to evaluate 3�3
�� to the nearest 38. __________________ten-thousandth.
Trigonometry Semester Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 561 Advanced Mathematical Concepts
Trigonometry Semester Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
39. Evaluate ��19����
12�
. 39. __________________
40. Use the Binomial Theorem to find the fourth term in 40. __________________the expansion of (2c � 3d)8.
41. Find the ordered triple that represents the vector 41. __________________from C(2, 0, 1) to D(6, 4, 3).
42. Find the distance between the lines with equations 42. __________________3x � 2y � 1 � 0 and 3x � 2y � 4 � 0.
43. Find an ordered pair to represent u� in 43. __________________u� � 2w� � v� if w� � 2, 3� and v� � 5, 5�.
44. Write the rectangular equation y � 1 in polar form. 44. __________________
45. Write the polar equation � � ��2� in rectangular form. 45. __________________
46. If CD� is a vector from C(1, 2, �1) to D(2, 3, 2), find 46. __________________the magnitude of CD�.
47. Are the vectors 2, �2, 1� and 3, 2, �2� perpendicular? 47. __________________Write yes or no.
48. Solve 235 � 4e0.35t. Round your answer to the 48. __________________nearest hundredth.
49. Write 0.6�3� as a fraction. 49. __________________
50. Graph the equation y � 3x�1. 50.
BLANK
© Glencoe/McGraw-Hill 563 Advanced Mathematical Concepts
Trigonometry Final Test
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right of each problem.
1. Given ƒ(x) � 2x � 4 and g(x) � x3 � x � 2, find ( ƒ � g)(x). 1. ________A. x3 � x � 2 B. x3 � 3x � 6C. �x3 � 3x � 2 D. �x3 � x � 2
2. Describe the graph of ƒ(x) � �xx2 �
�255�. 2. ________
A. The graph has infinite discontinuity.B. The graph has jump discontinuity.C. The graph has point discontinuity.D. The graph is continuous.
3. Choose the graph of the relation whose inverse is a function. 3. ________A. B. C. D.
4. Which is the graph of the system? x 0 4. ________y 02x � y � 2
A. B. C. D.
5. For which measures does � ABC have no solution? 5. ________A. A � 60°, a � 8, b � 8B. A � 70°, b � 10, c � 10C. A � 45°, a � 1, b � 20D. A � 60°, B � 30°, c � 2
© Glencoe/McGraw-Hill 564 Advanced Mathematical Concepts
6. Choose the amplitude, period, and phase shift of the function 6. ________y � �13� cos (2x � �).
A. �13�, �, ��2� B. �13�, �, � C. 3, �, ��2� D. �13�, �, �4��
7. Given ƒ(x) � x � 1 and g(x) � x2 � 2, find [ g ° ƒ ](x). 7. ________A. x2 � 2x � 3 B. x2 � 2x � 1 C. x2 � 3 D. x3 � x2 � 2x � 2
8. Find one positive angle and one negative angle that are coterminal 8. ________with the angle �
34��.
A. �54��, ��
114
�� B. �
23��, ��
54�� C. �
114
��, � �
�4� D. �
114
��, ��
54��
9. Simplify �1 �si
cno�s2 ��. 9. ________
A. �csoins
��� B. tan � C. sin � D. cos �
10. Find an ordered triple to represent u� in u� � 2v� � 3w� if v� � �1, 0, 2� 10. ________and w� � 2, 3, 1�.A. 2, 3, 1� B. 4, 3, 2� C. 3, 1, �2� D. 4, 9, 7�
11. Write log7 49 � 2 in exponential form. 11. ________A. 27 � 49 B. 72 � 49
C. 7�12
�� 49 D. 49
�12
�� 7
12. Write parametric equations of �4x � 5y � 10. 12. ________A. x � t; y � �45�t � 2 B. x � t; y � 4t � 10C. x � t; y � �4t � 5 D. x � t; y � �54�t � 2
13. Describe the transformation that relates the graph of y � 100x3 13. ________to the parent graph y � x3.A. The parent graph is ref lected over the line y � x.B. The parent graph is stretched horizontally.C. The parent graph is stretched vertically.D. The parent graph is translated horizontally right 1 unit.
14. Identify the polar form of the linear equation �3�x � y � 4. 14. ________A. 4 � r cos (� � 30°) B. 2 � r cos (� � 30°)C. 3 � r sin (� � 70°) D. 2 � r cos (� � 42°)
Trigonometry Final Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 565 Advanced Mathematical Concepts
Trigonometry Final Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
15. Which function is an even function? 15. ________A. y � x4 � x2 B. y � x3
C. y � x4 � x5 D. y � 2x4 � 3x
16. Find the area of � ABC to the nearest tenth if B � 30°, C � 70°, 16. ________and a � 10.A. 15.2 units2 B. 30.3 units2
C. 19.2 units2 D. 23.9 units2
17. The line with equation 9x � y � 2 is perpendicular to the line that 17. ________passes throughA. (0, 0) and (1, 3). B. (1, 2) and (10, 3).C. (2, �5) and (3, 4). D. (10, 4) and (1, 5).
18. Find the 17th term of the arithmetic sequence �13�, 1, �53�, . . . . 18. ________
A. 9 B. �137� C. 11 D. 7
19. Solve 2x2 � 4x � 1 � 0 by using the Quadratic Formula. 19. ________
A. 1 ��22�� B. �1 �
�22�� C. 2, 4 D. �1 �6�
20. What are the dimensions of matrix AB if A is a 2 � 3 matrix 20. ________and B is a 3 � 7 matrix?A. 7 � 2 B. 3 � 3 C. 3 � 2 D. 2 � 7
21. Find the polar coordinates of the point whose rectangular 21. ________coordinates are (�1, 1).
A. ��2�, ��4�� B. ��2�, ��
34���
C. ��2�, � ��4�� D. ��2�, �
34���
22. Find the fifteenth term of the geometric sequence �53�, �56�, �152�, . . . . 22. ________
A. �1965,608� B. �49,
5152� C. �98,
5304� D. �2
1�
23. Find the magnitude of AB� from A(2, 4, 0) to B(1, 2, 2). 23. ________A. �7� B. 2�3� C. 3 D. �5�
© Glencoe/McGraw-Hill 566 Advanced Mathematical Concepts
24. Choose the graph of y � csc x on the interval 0 � x � 2�. 24. ________A. B.
C. D.
25. Without graphing, describe the end behavior of y � �x4 � 2x3 � x2 � 1. 25. ________A. y → ∞ as x → ∞; y → ∞ as x → �∞B. y →�∞ as x → ∞; y → �∞ as x → �∞C. y →�∞ as x → ∞; y → ∞ as x → �∞D. y → ∞ as x → ∞; y → �∞ as x → �∞
26. Which expression is equivalent to sin (90° � �)? 26. ________A. sin � B. cos � C. tan � D. sec �
27. Describe the transformation that relates the graph of y � ��x� �� 3� 27. ________to the parent graph y � �x�.A. The parent graph is reflected over the x-axis and translated left 3 units.B. The parent graph is compressed vertically and translated right 3 units.C. The parent graph is reflected over the x-axis and translated up 3 units.D. The parent graph is reflected over the y-axis and translated left 3 units.
28. Express cos 854° as a function of an angle in Quandrant II. 28. ________A. tan 134° B. sin 64° C. cos 134° D. �cos 134°
29. Express �3
2�7�a�9b�2� by using rational exponents.
A. 3a3b�32� B. 9a6b�23
�C. 27a3b
�23
�D. 3a3b
�23
�29. ___________________________________
30. Identify the equation of the tangent function with period 2�, 30. ________phase shift �
�2�, and vertical shift �5.
A. y � tan (x � �) � 5 B. y � tan ��12�x � ��4�� � 5
C. y � �5 tan ��12�x � ��4�� D. y � tan �2x � �
�2��
Trigonometry Final Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 567 Advanced Mathematical Concepts
Trigonometry Final Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
31. Given ƒ(x) � x3 � 2x, find ƒ(�2). 31. __________________
32. Find the area of a sector if the central angle measures 32. __________________35° and the radius of the circle is 45 centimeters. Round to the nearest tenth.
33. Find A � B if A � � and B � � . 33. __________________
34. Solve the system of equations algebraically. 34. __________________3x � 2y � 3�2x � 5y � 17
35. Find the inverse of y � (x � 1)3. 35. __________________
36. State the domain and range of the relation 36. __________________{(�2, 5), (0, 3), (4, 5), (9, �3)}. Then state whether the relation is a function.
37. Graph y � �|x � 1| � 2. 37.
38. Find the sum of the infinite geometric 38. __________________series 35 � 7 � 1.4 � . . . .
39. Find AB if A � � and B � � . 39. __________________
40. Simplify sin � csc � (sin2 � � cos2 �). 40. __________________
41. A football is kicked with an initial velocity of 38 feet per 41. __________________second at an angle of 32° to the horizontal. How far has the football traveled horizontally after 0.25 second? Round your answer to the nearest tenth.
42. Find the sum of the first 23 terms of the arithmetic 42. __________________series �10 � �3 � 4 � . . . .
23
3�1
�20
31
�14
22
33
1�2
© Glencoe/McGraw-Hill 568 Advanced Mathematical Concepts
43. Graph the polar equation � � ��3�. 43.
44. During one year, the cost of tuition, room, and board at a 44. __________________state university increased 6%. If the cost continues to increase at a rate of 6% per year, how long will it take for the cost of tuition, room, and board to triple? Round your answer to the nearest year.
45. Name four different pairs 45. __________________of polar coordinates thatrepresent the point A.
46. Evaluate log7 15 to four decimal places. 46. __________________
47. Use a calculator to approximate the value of sec (�137°) 47. __________________to four decimal places.
48. Find the sum of the first 13 terms of the geometric series 48. __________________2 � 6 � 18 � . . . .
49. Find the exact value of sin 105°. 49. __________________
50. Find the multiplicative inverse of � . 50. __________________
51. Find the value of � �. 51. __________________�1
423
21
52
Trigonometry Final Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 569 Advanced Mathematical Concepts
Trigonometry Final Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
52. The vector u� has a magnitude of 4.5 centimeters and a 52. __________________direction of 56°. Find the magnitude of its horizontal component. Round to the nearest hundredth.
53. Graph the equation y � 2 sin (2x � �). 53.
54. Write 3�2 � �19� in logarithmic form. 54. __________________
55. Use a sum or difference identity to f ind the value of 55. __________________cos 285°.
56. Solve e�1x
�� 5. Round your answer to the nearest hundredth. 56. __________________
57. Write MN� as a sum of unit vectors for M(2, 1, 0) 57. __________________and N(5, �3, 2).
58. Express 3 � 5 � 9 � 17 � . . . � 513 using sigma notation. 58. __________________
59. Express 2 � 3i in polar form. Express your answer in 59. __________________radians to the nearest hundredth.
60. Find the third iterate of the function ƒ(z) � 2z � 1, if the 60. __________________initial value is z0 � 1 � i.
61. Find the rectangular coordinates of the point whose polar 61. __________________
coordinates are �2�2�, �54���.
62. Find the discriminant of w2 � 4w � 5 � 0 and describe the 62. __________________nature of the roots of the equation.
63. Find the value of Cos�1 ��12�� in degrees. 63. __________________
64. State the degree and leading coefficient of the polynomial 64. __________________function ƒ(x) � �6x3 � 2x4 � 3x5 � 2.
© Glencoe/McGraw-Hill 570 Advanced Mathematical Concepts
65. Write an equation in slope-intercept form of the line with 65. __________________the given parametric equations. x � �3t � 5
y � 1 � t
66. Solve the system of equations. 66. __________________x � 5y � z � 12x � y � 2z � 2x � 3y � 4z � 6
67. Use the Remainder Theorem to find the remainder when 67. __________________2x3 � x2 � 2x is divided by x � 1.
68. Find the value of cos �Sin�1 ���23���. 68. __________________
69. Solve log5 x � log5 9 � log5 4. 69. __________________
70. Find the rational zero(s) of ƒ(x) � 3x3 � 2x2 � 6x � 4. 70. __________________
71. Find the number of possible positive real zeros of 71. __________________ƒ(x) � 3x4 � 2x3 � 3x2 � x � 3.
72. List all possible rational zeros of ƒ(x) � 3x3 � bx2 � cx � 2, 72. __________________where b and c are integers.
73. Given a central angle of 87°, f ind the length of its 73. __________________intercepted arc in a circle of radius 19 centimeters.Round to the nearest tenth.
74. Solve � ABC if A � 27.2°, B � 76.1°, and a � 31.2. Round 74. __________________to the nearest tenth.
75. Solve � ABC if A � 40°, b � 10.2, and c � 5.7. Round to 75. __________________the nearest tenth.
Trigonometry Final Test (continued)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill A1 Advanced Mathematical Concepts
SAT and ACT Practice Answer Sheet(10 Questions)
NAME _____________________________ DATE _______________ PERIOD ________
0 0 0
.. ./ /
.
99 9 9
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
© Glencoe/McGraw-Hill A2 Advanced Mathematical Concepts
SAT and ACT Practice Answer Sheet(20 Questions)
NAME _____________________________ DATE _______________ PERIOD ________
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
Answers (Lesson 12-1)
© Glencoe/McGraw-Hill A3 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill51
1A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
12-1 n
2�
n�
1; 5
1, 7
0, 9
25 � 2
3 � 2 1;
n2
�n
1 � 21 � 2
n2
�n
� 1
1 � 21 � 2
Qu
ad
ratic
Fo
rmu
las
for
Se
qu
en
ce
sA
n o
rdin
ary
arit
hm
etic
seq
uen
ce is
for
med
usi
ng
a ru
le s
uch
as
bn�
c. T
he
firs
t te
rm is
c,
bis
cal
led
the
com
mon
dif
fere
nce
, an
d n
take
s on
th
e va
lues
0, 1
, 2, 3
, an
d so
on
. Th
e va
lue
of t
erm
n�
1eq
ual
s b
(n�
1)�
cor
bn
�b
�c.
So,
th
e va
lue
of a
ter
m is
a
fun
ctio
n o
f th
e te
rm n
um
ber.
Som
e se
quen
ces
use
qu
adra
tic
fun
ctio
ns.
Am
eth
od c
alle
d fi
nit
ed
iffe
ren
ces
can
be
use
d to
fin
d th
e va
lues
of
the
term
s. N
otic
e w
hat
hap
pen
s w
hen
you
su
btra
ct t
wic
e as
sh
own
in t
his
tab
le.
nan
2�
bn�
c0
ca
�b
1a
�b
�c
3a�
b2a
24a
�2b
�c
5a�
b2a
39a
�3b
�c
7a�
b2a
416
a�
4b�
cA
sequ
ence
th
at y
ield
s a
com
mon
dif
fere
nce
aft
er t
wo
subt
ract
ion
sca
n b
e ge
ner
ated
by
a qu
adra
tic
expr
essi
on. F
or e
xam
ple,
th
ese
quen
ce 1
, 5, 1
2, 2
2, 3
5, .
. . g
ives
a c
omm
on d
iffe
ren
ce o
f 3
afte
rtw
o su
btra
ctio
ns.
Usi
ng
the
tabl
e ab
ove,
you
wri
te a
nd
solv
e th
ree
equ
atio
ns
to f
ind
the
gen
eral
ru
le. T
he
equ
atio
ns
are
1�
c,5
�a
�b
�c,
an
d 12
�4a
�2b
�c.
S
olve
eac
h p
rob
lem
.
1.R
efer
to
the
sequ
ence
in t
he
exam
ple
abov
e. S
olve
th
e sy
stem
of
equ
atio
ns
for
a, b
, an
d c
and
then
fin
d th
e qu
adra
tic
expr
essi
onfo
r th
e se
quen
ce.
Th
en w
rite
th
e n
ext
thre
e te
rms.
2.T
he
nu
mbe
r of
lin
e se
gmen
ts c
onn
ecti
ng
npo
ints
for
ms
the
sequ
ence
0, 0
, 1, 3
, 6, 1
0, .
. . ,
in w
hic
h n
is t
he
nu
mbe
r of
poi
nts
and
the
term
val
ue
is t
he
nu
mbe
r of
lin
e se
gmen
ts. W
hat
is t
he
com
mon
dif
fere
nce
aft
er t
he
seco
nd
subt
ract
ion
? F
ind
a qu
adra
tic
expr
essi
on f
or t
he
term
val
ue.
3.T
he
max
imu
m n
um
ber
of r
egio
ns
form
ed b
y n
chor
ds in
a c
ircl
efo
rms
the
sequ
ence
1, 2
, 4, 7
, 11,
16,
. . .
(A
chor
d is
a li
ne
segm
ent
join
ing
any
two
poin
ts o
n a
cir
cle.
) D
raw
cir
cles
to
illu
stra
te t
he
firs
t fo
ur
term
s of
th
e se
quen
ce.
Th
en f
ind
a qu
adra
tic
expr
essi
onfo
r th
e te
rm v
alu
e.
© G
lenc
oe/M
cGra
w-H
ill51
0A
dva
nced
Mat
hem
atic
al C
once
pts
Ari
thm
etic
Se
qu
en
ce
s a
nd
Se
rie
sFi
nd
th
e n
ext
fou
r te
rms
in e
ach
ari
thm
etic
seq
uen
ce.
1.�
1.1,
0.6
, 2.3
, . .
.2.
16, 1
3, 1
0, .
. .3.
p, p
�2,
p�
4,. .
.4.
0, 5
.7, 7
.4, 9
.17,
4, 1
,�2
p�
6, p
�8,
p�
10, p
�12
For
exer
cise
s 4
–12,
ass
um
e th
at e
ach
seq
uen
ce o
r se
ries
is a
rith
met
ic.
4.F
ind
the
24th
ter
m in
th
e se
quen
ce f
or w
hic
h a
1�
�27
an
d d
�3.
42
5.F
ind
nfo
r th
e se
quen
ce f
or w
hic
h a
n�
27, a
1�
�12
, an
d d
�3.
14
6.F
ind
dfo
r th
e se
quen
ce f
or w
hic
h a
1�
�12
an
d a 23
�32
.2
7.W
hat
is t
he
firs
t te
rm in
th
e se
quen
ce f
or w
hic
h d
��
3 an
d a 6
�5?
20
8.W
hat
is t
he
firs
t te
rm in
th
e se
quen
ce f
or w
hic
h d
��
�1 3�an
d a 7
��
3?�
1
9.F
ind
the
6th
ter
m in
th
e se
quen
ce�
3�
�2�,
0, 3
��
2�, .
. . .
12�
4�2�
10.F
ind
the
45th
ter
m in
th
e se
quen
ce�
17,�
11,�
5, .
. ..
247
11.
Wri
te a
seq
uen
ce t
hat
has
th
ree
arit
hm
etic
mea
ns
betw
een
35
and
45.
35, 3
7.5,
40,
42.
5, 4
5
12.W
rite
a s
equ
ence
th
at h
as t
wo
arit
hm
etic
mea
ns
betw
een
�7
and
2.75
.�
7,�
3.75
,�0.
5, 2
.75
13.F
ind
the
sum
of
the
firs
t 13
ter
ms
in t
he
seri
es�
5�
1�
7�
. . .
�67
.40
3
14.F
ind
the
sum
of
the
firs
t 62
ter
ms
in t
he
seri
es�
23�
21.5
�20
�. .
. .
1410
.5
15.A
ud
itor
ium
Des
ign
Wak
efie
ld A
udi
tori
um
has
26
row
s, a
nd
the
firs
t ro
w h
as 2
2se
ats.
Th
e n
um
ber
of s
eats
in e
ach
row
incr
ease
s by
4 a
s yo
u m
ove
tow
ard
the
back
of t
he
audi
tori
um
. Wh
at is
th
e se
atin
g ca
paci
ty o
f th
is a
udi
tori
um
?18
72
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
12-1
Answers (Lesson 12-2)
© Glencoe/McGraw-Hill A4 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill51
4A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
12-2
Se
qu
en
ce
s a
s F
un
ctio
ns
Age
omet
ric
seq
uen
ceca
n b
e de
fin
ed a
s a
fun
ctio
n w
hos
e do
mai
n
is t
he
set
of p
osit
ive
inte
gers
.n
�1
23
4...
↓↓
↓↓
↓f(
n)�
ar1
– 1
ar2
– 1
ar3
– 1
ar4
– 1
...In
th
e ex
erci
ses,
you
wil
l hav
e th
e op
port
un
ity
to e
xplo
re g
eom
etri
cse
quen
ces
from
a f
un
ctio
n a
nd
grap
hin
g po
int
of v
iew
.
Gra
ph
eac
h g
eom
etri
c se
qu
ence
for
n =
1, 2
, 3 a
nd
4.
1.f(
n)�
2n2.
f(n
)�(0
.5)n
3.f(
n)�
(–2)
n4.
f(n
)�(–
0.5)
n
5.D
escr
ibe
how
th
e gr
aph
of
a ge
omet
ric
sequ
ence
dep
ends
on
th
eco
mm
on r
atio
.r
�1:
gra
ph
rise
s to
the
rig
ht.
r�
– 1:
gra
ph
rise
s an
d f
alls
and
hig
h an
d lo
w p
oin
ts
mo
ve a
way
fro
m t
he n
-axi
s.0
�r
�1:
gra
ph
falls
to
the
rig
ht.
– 1�
r�
0: g
rap
h ri
ses
and
fal
ls a
nd h
igh
and
low
po
ints
ap
pro
ach
the
n-a
xis.
6.L
et f
(n)�
2n ,
wh
ere
n is
a p
osit
ive
inte
ger.
a.S
how
gra
phic
ally
th
at f
or a
ny
Mth
e gr
aph
of
f(n
) ri
ses
abov
e an
d st
ays
abov
e th
e h
oriz
onta
l lin
e y
�M
.b
.S
how
alg
ebra
ical
ly t
hat
for
an
y M
, th
ere
is a
pos
itiv
e in
tege
r N
such
th
at 2
n�
Mfo
r al
l n
�N
.C
hoo
se a
po
siti
ve in
teg
er N
�lo
g2M
.T
hen
2N�
2log
2M�
M.
© G
lenc
oe/M
cGra
w-H
ill51
3A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Ge
om
etr
ic S
eq
ue
nc
es
an
d S
eri
es
Det
erm
ine
the
com
mon
rat
io a
nd
fin
d t
he
nex
t th
ree
term
s of
eac
h g
eom
etri
cse
qu
ence
.
1.�
1, 2
,�4,
. . .
2.�
4,�
3,�
�9 4� , .
. .3.
12,�
18, 2
7, .
. .
�2;
8,�
16, 3
2�3 4� ;
��2 17 6�
,��8 61 4�
,��2 24 53 6�
��3 2� ;
��8 21 �
, �24 43 �
, ��72 89 �
For
exer
cise
s 4
–9, a
ssu
me
that
eac
h s
equ
ence
or
seri
es is
geo
met
ric.
4.F
ind
the
fift
h t
erm
of
the
sequ
ence
20,
0.2
, 0.0
02, .
. . .
0.00
0000
2
5.F
ind
the
nin
th t
erm
of
the
sequ
ence
�3�,
�3,
3�
3�, .
. . .
81�
3�
6.If
r�
2 an
d a 4
�28
, fin
d th
e fi
rst
term
of
the
sequ
ence
. � 27 �
7.F
ind
the
firs
t th
ree
term
s of
th
e se
quen
ce f
or w
hic
h a
4�
8.4
and
r�
4.0.
1312
5, 0
.525
, 2.1
8.F
ind
the
firs
t th
ree
term
s of
th
e se
quen
ce f
or w
hic
h a
6�
� 31 2�an
d r
��1 2� .
1, �1 2� ,
� 41 �
9.W
rite
a s
equ
ence
th
at h
as t
wo
geom
etri
c m
ean
s be
twee
n 2
an
d 0.
25.
2, 1
, 0.5
, 0.2
5
10.W
rite
a s
equ
ence
th
at h
as t
hre
e ge
omet
ric
mea
ns
betw
een
�32
an
d�
2.�
32, �
16,�
8,�
4,�
2
11.
Fin
d th
e su
m o
f th
e fi
rst
eigh
t te
rms
of t
he
seri
es �3 4�
�� 29 0�
�� 12 07 0�
�. .
. .
abo
ut 1
.843
51
12.F
ind
the
sum
of
the
firs
t 10
ter
ms
of t
he
seri
es�
3�
12�
48�
. . .
.62
9,14
5
13.P
opu
lati
on G
row
thA
city
of
100,
000
peop
le is
gro
win
g at
a r
ate
of
5.2%
per
yea
r. A
ssu
min
g th
is g
row
th r
ate
rem
ain
s co
nst
ant,
est
imat
e th
e po
pula
tion
of
the
city
5 y
ears
fro
m n
ow.
abo
ut 1
28,8
48
12-2
Answers (Lesson 12-3)
© Glencoe/McGraw-Hill A5 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill51
7A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
12-3
So
lvin
g E
qu
atio
ns
Usi
ng
Se
qu
en
ce
sYo
u c
an u
se s
equ
ence
s to
sol
ve m
any
equ
atio
ns.
For
exa
mpl
e,
con
side
r x2
�x
– 1
�0.
You
can
pro
ceed
as
foll
ows.
x2�
x –
1�
0
x(x
�1)
�1
x�
Nex
t, d
efin
e th
e se
quen
ce: a
1�
0 an
d a n
�.
Th
e li
mit
of
the
sequ
ence
is a
sol
uti
on t
o th
e or
igin
al e
quat
ion
.
1.L
et a
1�
0 an
d a n
�.
a.W
rite
th
e fi
rst
five
ter
ms
of t
he
sequ
ence
. Do
not
sim
plif
y.
1
11
1__
___
__
___
____
____
___
____
____
____
____
_0,
1 +
0 ,
1 +
1 ,
1 +
1
,1
+
1__
___
____
____
___
1 +
11
+1
____
__1
+ 1
b.
Wri
te d
ecim
als
for
the
firs
t fi
ve t
erm
s of
th
e se
quen
ce.
0, 1
, 0.5
, 0.6
667,
0.6
c.U
se a
cal
cula
tor
to c
ompu
te a
6, a
7, a
8, a
nd
a 9. C
ompa
re a
9w
ith
the
posi
tive
sol
uti
on o
f x2
�x
– 1
�0
fou
nd
by u
sin
g th
e qu
adra
tic
form
ula
.0.
625,
0.6
154,
0.6
190,
0.6
176;
so
luti
on
by
qua
dra
tic
form
ula:
0.6
180
2.U
se t
he
met
hod
des
crib
ed a
bove
to
fin
d a
root
of
3x2
–2x
–3
�0.
–0.7
208
3.W
rite
a B
AS
IC p
rogr
am u
sin
g th
e pr
oced
ure
ou
tlin
ed a
bove
to
fin
d a
root
of
the
equ
atio
n 3
x2–
2x–
3�
0. I
n t
he
prog
ram
,
let
a 1�
0 an
d a n
�.
Ru
n t
he
prog
ram
. Com
pare
th
e
tim
e it
tak
es t
o ru
n t
he
prog
ram
to
the
tim
e it
tak
es t
o ev
alu
ate
the
term
s of
th
e se
quen
ce b
y u
sin
g a
calc
ula
tor.
10 D
IM A
[100
]20
A[1
] �
030
FO
R I
�2
TO
100
40 A
[I]
�3/
(3*A
[I �
1] �
2)50
NE
XT
I60
PR
INT
A[1
00]3
��
3an
�1�
2
1�
�1
� a
n�
1
1�
�1
� a
n�
1
1� 1
�x
© G
lenc
oe/M
cGra
w-H
ill51
6A
dva
nced
Mat
hem
atic
al C
once
pts
Infi
nite
Se
qu
en
ce
an
d S
eri
es
Fin
d e
ach
lim
it, o
r st
ate
that
th
e lim
it d
oes
not
exi
st a
nd
exp
lain
you
r re
ason
ing
.1.
lim
n→
∞� nn 22
��11
�2.
lim
n→
∞�4 3n n2 2� �
5 4n�
1� 34 �
3.li
mn
→∞�5n
62
n�1
�4.
lim
n→
∞�(n
�1 5)( n3 2n
�1)
�
do
es n
ot
exis
t; d
ivid
ing
by
n2 ,
� 53 �
we
find
lim
n→
∞, w
hich
sim
plif
ies
to �5 6� �
00�
��5 0�
as n
app
roac
hes
infin
ity.
Sin
ce t
his
frac
tio
n is
und
efin
ed, t
he
seq
uenc
e ha
s no
lim
it.
5.li
mn
→∞�3n
� 4n(� 2
1)n
�6.
lim
n→
∞�n
3 n� 21
�
0d
oes
no
t ex
ist;
lim
n→
∞�n
3 n� 21
��
lim n→
∞�n
�� n1 2�
�.lim n→
∞� n1 2�
�0,
but
as
n
app
roac
hes
infin
ity,
nb
eco
mes
in
crea
sing
ly la
rge,
so
the
se
que
nce
has
no li
mit
.
Wri
te e
ach
rep
eati
ng
dec
imal
as
a fr
acti
on.
7.0.
7�5�8.
0.5�9�
2��2 35 3�
� 21 76 �
Fin
d t
he
sum
of
each
infi
nit
e se
ries
, or
stat
e th
at t
he
sum
doe
s n
ot e
xist
an
d
exp
lain
you
r re
ason
ing
.
9.�2 5�
�� 26 5�
�� 11 28 5�
�. .
.10
.�3 4�
��1 85 �
��7 15 6�
�. .
.
1d
oes
no
t ex
ist;
sin
ce r
��5 2� ,
�r��
1 an
d
the
seq
uenc
e in
crea
ses
wit
hout
lim
it.
11.
Ph
ysic
sA
ten
nis
bal
l is
drop
ped
from
a h
eigh
t of
55
feet
an
dbo
un
ces
�3 5�of
th
e di
stan
ce a
fter
eac
h f
all.
a.F
ind
the
firs
t se
ven
ter
ms
of t
he
infi
nit
e se
ries
rep
rese
nti
ng
the
vert
ical
dis
tan
ces
trav
eled
by
the
ball
.55
, 33,
33,
19.8
, 19.
8, 1
1.88
, 11.
88b
.W
hat
is t
he
tota
l ver
tica
l dis
tan
ce t
he
ball
tra
vels
bef
ore
com
ing
to r
est?
220
ft
5�
� n1 2��
� n6 �
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
12-3
© G
lenc
oe/M
cGra
w-H
ill52
0A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
12-4
Alt
ern
atin
g S
eri
es
Th
e se
ries
bel
ow is
cal
led
an a
lter
nat
ing
seri
es.
1 –
1�
1 –
1�
...
Th
e re
ason
is t
hat
th
e si
gns
of t
he
term
s al
tern
ate.
An
inte
rest
ing
ques
tion
is w
het
her
th
e se
ries
con
verg
es. I
n t
he
exer
cise
s, y
ou w
ill
hav
e an
opp
ortu
nit
y to
exp
lore
th
is s
erie
s an
d ot
her
s li
ke it
.
1.C
onsi
der
1 –
1�
1 –
1�
....
a.W
rite
an
arg
um
ent
that
su
gges
ts t
hat
th
e su
m is
1.
1�
1�
1�
1�
...�
1�
(–1
�1)
�(–
1�
1 )�
...�
1�
0�
0�
...�
1b
.W
rite
an
arg
um
ent
that
su
gges
ts t
hat
th
e su
m is
0.
1�
1�
1�
1�
...�
(1�
1)�
(1�
1)�
(1�
1)�
...�
0�
0�
0�
...�
0c.
Wri
te a
n a
rgu
men
t th
at s
ugg
ests
th
at t
her
e is
no
sum
. (H
int:
Con
side
r th
e se
quen
ce o
f pa
rtia
l su
ms.
)Le
t S
nb
e th
e n
th p
arti
al s
um.
The
n
S n�
�S
ince
1, 0
, 1, 0
, ...
has
no li
mit
, the
ori
gin
alse
ries
has
no
sum
.
If t
he
seri
es f
orm
ed b
y ta
kin
g th
e ab
solu
te v
alu
es o
f th
e te
rms
of a
give
n s
erie
s is
con
verg
ent,
th
en t
he
give
n s
erie
s is
sai
d to
be
abso
lute
ly c
onve
rgen
t. I
t ca
n b
e sh
own
th
at a
ny
abso
lute
ly
con
verg
ent
seri
es is
con
verg
ent.
2.M
ake
up
an a
lter
nat
ing
seri
es, o
ther
th
an a
geo
met
ric
seri
es w
ith
neg
ativ
e co
mm
on r
atio
, th
at h
as a
su
m. J
ust
ify
you
r an
swer
.S
amp
le a
nsw
er:
��
��
...is
co
nver
gen
t b
ecau
se
the
abso
lute
val
ue o
f ea
ch t
erm
is le
ss t
han
or
equa
l to
the
co
rres
po
ndin
g t
erm
in a
p-s
erie
sw
ith
p�
2.
1 � 161 � 9
1 � 41 � 1
1 if
nis
od
d.
0 if
nis
eve
n.
© G
lenc
oe/M
cGra
w-H
ill51
9A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Co
nve
rge
nt
an
d D
ive
rge
nt
Se
rie
s
Use
th
e ra
tio
test
to
det
erm
ine
wh
eth
er e
ach
ser
ies
isco
nve
rgen
t or
div
erg
ent.
1.�1 2�
��2 22 2�
��3 22 3�
��4 22 4�
�. .
.2.
0.00
6�
0.06
�0.
6�
. . .
conv
erg
ent
div
erg
ent
3.� 1
�
4 2�
3�
�� 1
�2
8 �3
�4
��� 1
�2
�1 36 �4
�5
��
. . .
conv
erg
ent
4.5
�� 35 3�
�� 55 3�
�� 75 3�
�. .
.
div
erg
ent
Use
th
e co
mp
aris
on t
est
to d
eter
min
e w
het
her
eac
h s
erie
s is
con
verg
ent
or d
iver
gen
t.5.
2�
� 22 3��
� 32 3��
� 42 3��
. . .
6.�5 2�
�1
��5 8�
�� 15 1�
�. .
.
conv
erg
ent
div
erg
ent
7.E
colo
gyA
lan
dfil
l is
leak
ing
a to
xic
chem
ical
. Six
mon
ths
afte
rth
e le
ak w
as d
etec
ted,
th
e ch
emic
al h
ad s
prea
d 12
50 m
eter
s fr
omth
e la
ndf
ill.
Aft
er o
ne
year
, th
e ch
emic
al h
ad s
prea
d 50
0 m
eter
sm
ore,
an
d by
th
e en
d of
18
mon
ths,
it h
ad r
each
ed a
n a
ddit
ion
al20
0 m
eter
s.a.
If t
his
pat
tern
con
tin
ues
, how
far
wil
l th
e ch
emic
al s
prea
dfr
om t
he
lan
dfil
l aft
er 3
yea
rs?
abo
ut 2
075
m
b.
Wil
l th
e ch
emic
al e
ver
reac
h t
he
grou
nds
of
a h
ospi
tal l
ocat
ed25
00 m
eter
s aw
ay f
rom
th
e la
ndf
ill?
Exp
lain
.N
o, t
he s
um o
f th
e in
finit
e se
ries
mo
del
ing
thi
s si
tuat
ion
is a
bo
ut 2
083.
Thu
s th
e ch
emic
al w
illsp
read
no
mo
re t
han
abo
ut 2
083
met
ers.
12-4
Answers (Lesson 12-4)
© Glencoe/McGraw-Hill A6 Advanced Mathematical Concepts
Answers (Lesson 12-5)
© Glencoe/McGraw-Hill A7 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill52
3A
dva
nced
Mat
hem
atic
al C
once
pts
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
Str
ee
t N
etw
ork
s: F
ind
ing
All P
oss
ible
Ro
ute
sA
sect
ion
of
a ci
ty is
laid
ou
t in
squ
are
bloc
ks.
Goi
ng
nor
th f
rom
th
e in
ters
ecti
on o
f 1s
t A
ven
ue
and
1st
Str
eet,
th
e av
enu
es a
re
1st,
2n
d, 3
rd, a
nd
so o
n.
Goi
ng
east
, th
e st
reet
s ar
e n
um
bere
d in
th
e sa
me
way
.
Fac
tori
als
can
be
use
d to
fin
d th
e n
um
ber,
r(
e, n
), o
f di
ffer
ent
rou
tes
betw
een
tw
o in
ters
ecti
ons.
Th
e n
um
ber
of s
tree
ts g
oin
g ea
st is
e; t
he
nu
mbe
r of
ave
nu
es g
oin
gn
orth
is n
.
Th
e fo
llow
ing
prob
lem
s ex
amin
e th
e po
ssib
le r
oute
s fr
om o
ne
loca
tion
to
anot
her
. A
ssu
me
that
you
nev
er u
se a
rou
te t
hat
is
un
nec
essa
rily
lon
g. A
ssu
me
that
e�
1 an
d n
�1.
Sol
ve e
ach
pro
ble
m.
1.L
ist
all t
he
poss
ible
rou
tes
from
1st
Str
eet
and
1st
Ave
nu
e to
4t
h S
tree
t an
d 3r
d A
ven
ue.
Use
ord
ered
pai
rs t
o sh
ow t
he
rou
tes,
w
ith
str
eet
nu
mbe
rs f
irst
an
d av
enu
e n
um
bers
sec
ond.
Eac
h
rou
te m
ust
sta
rt a
t (1
, 1)
and
end
at (
4, 3
).(1
, 1) →
(2, 1
) → (3
, 1) →
(4, 1
) → (4
, 2) →
(4, 3
)(1
, 1) →
(2, 1
) → (3
, 1) →
(3, 2
) → (4
, 2) →
(4, 3
)(1
, 1) →
(2, 1
) → (3
, 1) →
(3, 2
) → (3
, 3) →
(4, 3
)(1
, 1) →
(2, 1
) → (2
, 2) →
(3, 2
) → (4
, 2) →
(4, 3
)(1
, 1) →
(2, 1
) → (2
, 2) →
(3, 2
) → (3
, 3) →
(4, 3
)(1
, 1) →
(2, 1
) → (2
, 2) →
(2, 3
) → (3
, 3) →
(4, 3
)(1
, 1) →
(1, 2
) → (2
, 2) →
(3, 2
) → (4
, 2) →
(4, 3
)(1
, 1) →
(1, 2
) → (2
, 2) →
(3, 2
) → (3
, 3) →
(4, 3
)(1
, 1) →
(1, 2
) → (2
, 2) →
(2, 3
) → (3
, 3) →
(4, 3
)(1
, 1) →
(1, 2
) → (1
, 3) →
(2, 3
) → (3
, 3) →
(4, 3
)2.
Use
the
form
ula
toco
mpu
teth
en
um
ber
ofro
ute
sfr
om (
1,1)
to(4
,3).
Th
ere
are
4 st
reet
s go
ing
east
an
d 3
aven
ues
goi
ng
nor
th.
�10
3.F
ind
the
nu
mbe
r of
rou
tes
from
1st
Str
eet
and
1st
Ave
nu
e to
7th
Str
eet
and
6th
Ave
nu
e.
�46
2(6
�5)
!�
6!5!
(3�
2)!
�3!
2!
Enr
ichm
ent
12-5
r(e,
n)
�[(
e�
1)�
(n�
1)]!
��
�(e
�1)
! (n
�1)
!
© G
lenc
oe/M
cGra
w-H
ill52
2A
dva
nced
Mat
hem
atic
al C
once
pts
Sig
ma
No
tatio
n a
nd
th
e n
th T
erm
Wri
te e
ach
exp
ress
ion
in e
xpan
ded
for
m a
nd
th
en f
ind
th
e su
m.
1.�5
n�
3(n
2�
2n)
2.�4
q�
1� q2 �
(32
�23 )
�(4
2�
24 )�
(52
�25 )
;�6
�2 1��
�2 2��
�2 3��
�2 4� ; �2 65 �
3.�5
t�1t(
t �
1)
4.�3
t�0(2
t�3)
1(1
�1)
�2(
2�
1)�
3(3
�1)
�[2
(0)�
3]�
[2(1
)�3]
�4(
4�
1)�
5(5
�1)
; 40
[2(2
)�3]
�[2
(3)�
3]; 0
5.�5
c�2(c
�2)
26.
�∞ i�110
��1 2� �i
(2�
2)2
�(3
�2)
2�
10��1 2� �1
�10
��1 2� �2�
(4�
2)2
�(5
�2)
2 ; 1
410
��1 2� �3�
. . .
�10
��1 2� ��; 1
0
Exp
ress
eac
h s
erie
s u
sin
g s
igm
a n
otat
ion
.
7.3
�6
�9
�12
�15
8.6
�24
�12
0�
. . .
�40
,320
�5
n�
13n
�8
n�
3n
!
9.�1 1�
��1 4�
��1 9�
�. .
.�
� 101 0�
10.
24�
19�
14�
. . .
�(�
1)
�10
n�
1� n1 2�
�5
n�
0(2
4�
5n)
11.
Sa
vin
gsK
ath
ryn
sta
rted
sav
ing
quar
ters
in a
jar.
Sh
e be
gan
by p
utt
ing
two
quar
ters
in t
he
jar
the
firs
t da
y an
d th
en s
he
incr
ease
d th
e n
um
ber
of q
uar
ters
sh
e pu
t in
th
e ja
r by
on
e ad
diti
onal
qu
arte
r ea
ch s
ucc
essi
ve d
ay.
a.U
se s
igm
a n
otat
ion
to
repr
esen
t th
e to
tal n
um
ber
of q
uar
ters
Kat
hry
n h
ad a
fter
30
days
.
b.
Fin
d th
e su
m r
epre
sen
ted
in p
art
a.
495
qua
rter
s
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
12-5
�30
n�
1(n�
1)
Answers (Lesson 12-6)
© Glencoe/McGraw-Hill A8 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill52
6A
dva
nced
Mat
hem
atic
al C
once
pts
Pa
tte
rns
in P
asc
al’s
Tri
an
gle
You
hav
e le
arn
ed t
hat
th
e co
effi
cien
ts in
th
e ex
pan
sion
of
(x
�y)
n
yiel
d a
nu
mbe
r py
ram
id c
alle
d P
asca
l’s t
rian
gle.
Row
1
Row
2
Row
3
Row
4
Row
5
Row
6
Row
7
As
man
y ro
ws
can
be
adde
d to
th
e bo
ttom
of
the
pyra
mid
as
you
nee
d.T
his
act
ivit
y ex
plor
es s
ome
of t
he
inte
rest
ing
prop
erti
es o
f th
is
fam
ous
nu
mbe
r py
ram
id.
1.P
ick
a ro
w o
f P
asca
l’s t
rian
gle.
a.W
hat
is t
he
sum
of
all t
he
nu
mbe
rs in
all
th
e ro
ws
abov
e th
e ro
w y
ou p
icke
d?S
ee s
tud
ents
’ wo
rk.
b.W
hat
is t
he
sum
of
all t
he
nu
mbe
rs in
th
e ro
w y
ou p
icke
d?S
ee s
tud
ents
’ wo
rk.
c.H
ow a
re y
our
answ
ers
for
part
s a
and
bre
late
d?T
he a
nsw
er f
or
par
t b
is 1
mo
re t
han
the
answ
er f
or
par
t a.
d.R
epea
t pa
rts
ath
rou
gh c
for
at le
ast
thre
e m
ore
row
s of
P
asca
l’s t
rian
gle.
Wh
at g
ener
aliz
atio
n s
eem
s to
be
tru
e?It
app
ears
that
the
sum
of t
he n
umbe
rs in
any
row
is 1
mor
eth
an th
e su
m o
f the
num
bers
in a
ll of
the
row
s ab
ove
it.e.
See
if y
ou c
an p
rove
you
r ge
ner
aliz
atio
n.
Sum
of n
umbe
rs in
row
n�
2n�
1 ; Th
e su
m o
f the
num
bers
in th
ero
ws
abov
e ro
w n
is 2
0+
21+
22+
. . .
+ 2n
�2 ,
whi
ch, b
y th
e fo
rmul
a fo
r the
sum
of a
geo
met
ric s
erie
s, is
2n
�1
�1.
2.P
ick
any
row
of
Pas
cal’s
tri
angl
e th
at c
omes
aft
er t
he
firs
t.a.
Sta
rtin
g at
th
e le
ft e
nd
of t
he
row
, fin
d th
e su
m o
f th
e od
d n
um
bere
d te
rms.
See
stu
den
ts’ w
ork
.b
.In
th
e sa
me
row
, fin
d th
e su
m o
f th
e ev
en n
um
bere
d te
rms.
See
stu
den
ts’ w
ork
.c.
How
do
the
sum
s in
par
ts a
an
d b
com
pare
?T
he s
ums
are
equa
l.d
.Rep
eat
part
s a
thro
ugh
cfo
r at
leas
t th
ree
oth
er r
ows
ofIn
any
ro
w o
fP
asca
l’s t
rian
gle.
Wh
at g
ener
aliz
atio
n s
eem
s to
be
tru
e?P
asca
l’s t
rian
gle
afte
r th
e fir
st, t
he s
um o
f th
e o
dd
num
ber
ed t
erm
s is
eq
ual t
o t
hesu
m o
f th
e ev
en n
umb
ered
ter
ms.
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
12-6
→ → → → → → →
1
11
12
1
13
31
14
64
1
15
1010
51
16
1520
156
1
© G
lenc
oe/M
cGra
w-H
ill52
5A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Th
e B
ino
mia
l Th
eo
rem
Use
Pas
cal’s
tri
ang
le t
o ex
pan
d e
ach
bin
omia
l.
1.(r
�3)
52.
(3a
�b)
4
r5�
15r4
�90
r3�
270r
2�
405r
�24
381
a4
�10
8a3 b
�54
a2 b
2�
12ab
3�
b4
Use
th
e B
inom
ial T
heo
rem
to
exp
and
eac
h b
inom
ial.
3.(x
�5)
44.
(3x
�2y
)4
x4
�20
x3
�15
0x2
�50
0x�
625
81x
4�
216x
3 y�
216x
2 y2
�96
xy3
�16
y4
5.(a
��
2�)5
6.(2
p�
3q)6
a5�
5�2�a
4�
20a3
�20
�2�a
2�
20a
�4�
2�64
p6
�57
6p5 q
�21
60p
4 q2
�43
20p
3 q3
�48
60p
2 q4
�29
16p
q5
�72
9q6
Fin
d t
he
des
ign
ated
ter
m o
f ea
ch b
inom
ial e
xpan
sion
.7.
4th
ter
m o
f (2
n�
3m)4
8.5t
h t
erm
of
(4a
�2b
)8
�21
6nm
328
6,72
0a4 b
4
9.6t
h t
erm
of
(3p
�q)
910
.3r
d te
rm o
f (a
�2�
3�)6
10,2
06p
4 q5
180a
4
11.
Ava
rsit
y vo
lley
ball
tea
m n
eeds
nin
e m
embe
rs. O
f th
ese
nin
em
embe
rs, a
t le
ast
five
mu
st b
e se
nio
rs. H
ow m
any
of t
he
poss
ible
gro
ups
of
jun
iors
an
d se
nio
rs h
ave
at le
ast
five
sen
iors
?25
6
12-6
Answers (Lesson 12-7)
© Glencoe/McGraw-Hill A9 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill52
9A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
Po
we
r S
eri
es
Ap
ower
ser
ies
is a
ser
ies
of t
he
form
a 0�
a 1x�
a 2x2
�a 3x
3�
...
wh
ere
each
aiis
a r
eal n
um
ber.
Man
y fu
nct
ion
s ca
n b
e re
pres
ente
d by
pow
er s
erie
s. F
or in
stan
ce, t
he
fun
ctio
n f
(x)�
exca
n b
e re
pres
ente
d by
th
e se
ries
ex�
1�
x�
��
....
Che
ck s
tud
ents
' gra
phs
.U
se a
gra
ph
ing
cal
cula
tor
or c
omp
ute
r to
gra
ph
th
e fu
nct
ion
s in
Exe
rcie
s 1–
4.
1.f 2
(x)�
1�
x2.
f 3(x
)�1
�x
�
3.f 4
(x)�
1�
x�
�4.
f 5(x
)�1
�x
��
�
5.W
rite
a s
tate
men
t th
at r
elat
es t
he
sequ
ence
of
grap
hs
sugg
este
d by
Exe
rcie
s 1–
4 an
d th
e fu
nct
ion
y�
ex .T
he f
unct
ions
def
ined
by
the
par
tial
sum
s co
nver
ge
to y
�ex .
Tha
t is
, as
nin
crea
ses,
the
grap
hs o
f fn
com
e in
to c
lose
r co
inci
denc
e w
ithth
e g
rap
h o
f y
�ex
for
mo
re a
nd m
ore
val
ues
of
x.
6.T
he
seri
es 1
�x2
�x4
�x6
�...
is a
pow
er s
erie
s fo
r w
hic
h e
ach
a i
�1.
Th
e se
ries
is a
lso
a ge
omet
ric
seri
es w
ith
fir
st t
erm
1 a
nd
com
mon
rat
io x
2 .a.
Fin
d th
e fu
nct
ion
th
at t
his
pow
er s
erie
s re
pres
ents
.y
�
b.
For
wh
at v
alu
es o
f x
does
th
e se
ries
giv
e th
e va
lues
of
the
fun
ctio
n in
par
t a?
–1�
x�
1
7.F
ind
a po
wer
ser
ies
repr
esen
tati
on f
or t
he
fun
ctio
n f
(x)�
.3
�3x
2�
3x4
�3x
6�
...3
� 1�
x2
1� 1
�x2
x4
� 4!
x3
� 3!
x2
� 2!
x3
� 3!
x2
� 2!
x2
� 2!
x3
� 3!
x2
� 2!
12-7
© G
lenc
oe/M
cGra
w-H
ill52
8A
dva
nced
Mat
hem
atic
al C
once
pts
Sp
ec
ial
Se
qu
en
ce
s a
nd
Se
rie
s
Fin
d e
ach
val
ue
to f
our
dec
imal
pla
ces.
1.ln
(�5)
2.ln
(�5.
7)3.
ln(�
1000
)i�
�1.
6094
i��
1.74
05i�
�6.
9078
Use
th
e fi
rst
five
ter
ms
of t
he
exp
onen
tial
ser
ies
and
a c
alcu
lato
rto
ap
pro
xim
ate
each
val
ue
to t
he
nea
rest
hu
nd
red
th.
4.e0.
55.
e1.2
1.65
3.29
6.e2.
77.
e0.9
12.8
42.
45
Use
th
e fi
rst
five
ter
ms
of t
he
trig
onom
etri
c se
ries
to
app
roxi
mat
eth
e va
lue
of e
ach
fu
nct
ion
to
fou
r d
ecim
al p
lace
s. T
hen
, com
par
eth
e ap
pro
xim
atio
n t
o th
e ac
tual
val
ue.
8.si
n �5 6� �
9.co
s �3 4� �
0.50
09; 0
.5�
0.70
57; �
0.70
71
Wri
te e
ach
com
ple
x n
um
ber
in e
xpon
enti
al f
orm
.
10.1
3 �cos
�� 3��
isi
n �� 3� �
11.
5�
5i
13ei�� 3�
5�2�e
i� 4� �
12.1
��
3�i13
.�
7�
7�3�i
2ei�� 3�
14ei�5 3� �
14.S
avi
ngs
Der
ika
depo
site
d $5
00 in
a s
avin
gs a
ccou
nt
wit
h a
4.5%
inte
rest
rat
e co
mpo
un
ded
con
tin
uou
sly.
(H
int:
Th
e fo
rmu
lafo
r co
nti
nu
ousl
y co
mpo
un
ded
inte
rest
is A
�P
ert.)
a.A
ppro
xim
ate
Der
ika’
s sa
vin
gs a
ccou
nt
bala
nce
aft
er 1
2 ye
ars
usi
ng
the
firs
t fo
ur
term
s of
th
e ex
pon
enti
al s
erie
s.ap
pro
xim
atel
y $8
56.0
2
b.
How
lon
g w
ill i
t ta
ke f
or D
erik
a’s
depo
sit
to d
oubl
e, p
rovi
ded
she
does
not
dep
osit
an
y ad
diti
onal
fu
nds
into
her
acc
oun
t?ab
out
15.
4 ye
ars
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
12-7
Answers (Lesson 12-8)
© Glencoe/McGraw-Hill A10 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill53
2A
dva
nced
Mat
hem
atic
al C
once
pts
De
pre
cia
tio
nTo
ru
n a
bu
sin
ess,
a c
ompa
ny
purc
has
es a
sset
s su
ch a
s eq
uip
men
t or
buil
din
gs. F
or t
ax p
urp
oses
, th
e co
mpa
ny
dist
ribu
tes
the
cost
of
thes
e as
sets
as
a bu
sin
ess
expe
nse
ove
r th
e c
ours
e of
a n
um
ber
ofye
ars.
Sin
ce a
sset
s de
prec
iate
(lo
se s
ome
of t
hei
r m
arke
t va
lue)
as
they
get
old
er, c
ompa
nie
s m
ust
be
able
to
figu
re t
he
depr
ecia
tion
expe
nse
th
ey a
re a
llow
ed t
o ta
ke w
hen
th
ey f
ile
thei
r in
com
e ta
xes.
Dep
reci
atio
n e
xpen
se is
a f
un
ctio
n o
f th
ese
thre
e va
lues
:1.
asse
t co
st, o
r th
e am
oun
t th
e co
mpa
ny
paid
for
th
e as
set;
2.es
tim
ated
use
ful
life
, or
the
nu
mbe
r of
yea
rs t
he
com
pan
y ca
n
expe
ct t
o u
se t
he
asse
t;3.
resi
du
al o
r tr
ade-
in v
alu
e, o
r th
e ex
pect
ed c
ash
val
ue
of t
he
asse
t at
th
e en
d of
its
use
ful l
ife.
In a
ny
give
n y
ear,
th
e b
ook
val
ue
of a
n a
sset
is e
qual
to
the
asse
t co
st m
inu
s th
e ac
cum
ula
ted
depr
ecia
tion
. Th
is v
alu
e re
pres
ents
th
eu
nu
sed
amou
nt
of a
sset
cos
t th
at t
he
com
pan
y m
ay d
epre
ciat
e in
futu
re y
ears
. T
he
use
ful l
ife
of t
he
asse
t is
ove
r on
ce it
s bo
ok v
alu
eis
equ
al t
o it
s re
sidu
al v
alu
e.
Th
ere
are
seve
ral m
eth
ods
of d
eter
min
ing
the
amou
nt
of
depr
ecia
tion
in a
giv
en y
ear.
In
th
e d
ecli
nin
g-b
alan
ce m
eth
od, t
he
depr
ecia
tion
exp
ense
all
owed
eac
h y
ear
is e
qual
to
the
book
val
ue
ofth
e as
set
at t
he
begi
nn
ing
of t
he
year
tim
es t
he
depr
ecia
tion
rat
e.S
ince
th
e de
prec
iati
on e
xpen
se f
or a
ny
year
is d
epen
den
t u
pon
th
ede
prec
iati
on e
xpen
se f
or t
he
prev
iou
s ye
ar, t
he
proc
ess
of
dete
rmin
ing
the
depr
ecia
tion
exp
ense
for
a y
ear
is a
n it
erat
ion
.
Th
e ta
ble
bel
ow s
how
s th
e fi
rst
two
iter
ates
of
the
dep
reci
atio
n s
ched
ule
for
a
$250
0 co
mp
ute
r w
ith
a r
esid
ual
val
ue
of $
500
if t
he
dep
reci
atio
n r
ate
is 4
0%.
1.F
ind
the
nex
t tw
o it
erat
es f
or t
he
depr
ecia
tion
exp
ense
fu
nct
ion
.$3
60; $
40 (o
nly
$40
rem
ains
of
the
asse
t co
st b
efo
re t
he
resi
dua
l val
ue is
rea
ched
)2.
Fin
d th
e n
ext
two
iter
ates
for
th
e en
d-of
-yea
r bo
ok v
alu
e fu
nct
ion
.$5
40; $
500
3.E
xpla
in t
he
depr
ecia
tion
exp
ense
for
yea
r 5.
The
re is
no
dep
reci
atio
n ex
pen
se b
ecau
se t
he b
oo
k va
lue
equa
ls t
he r
esid
ual v
alue
.
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
12-8 E
nd o
fA
sset
Dep
reci
atio
nB
oo
k Va
lue
atYe
arC
ost
Exp
ense
End
of
Year
1$2
500
$100
0$1
500
(40%
of $
2500
)($
2500
- $
1000
)
2$2
500
$600
$900
(40%
of $
1500
)($
1500
- $
600)
© G
lenc
oe/M
cGra
w-H
ill53
1A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Se
qu
en
ce
s a
nd
Ite
ratio
n
Fin
d t
he
firs
t fo
ur
iter
ates
of
each
fu
nct
ion
usi
ng
th
e g
iven
init
ial
valu
e. If
nec
essa
ry, r
oun
d y
our
answ
ers
to t
he
nea
rest
hu
nd
red
th.
1.ƒ(
x)�
x2�
4; x
0�
12.
ƒ(x)
�3x
�5;
x0
��
15,
29,
845
, 714
,029
2, 1
1, 3
8, 1
19
3.ƒ(
x)�
x2�
2; x
0�
�2
4.ƒ(
x)�
x(2.
5�
x); x
0�
32,
2, 2
, 2�
1.5,
�6,
�51
,�27
28.5
Fin
d t
he
firs
t th
ree
iter
ates
of
the
fun
ctio
n ƒ
(z)�
2z
�(3
�i)
for
each
init
ial v
alu
e.
5.z 0
�i
6.z 0
�3
�i
�3
�i
3�
3i�
9�
i3
�7i
�21
�i
3�
15i
7.z 0
�0.
5�
i8.
z 0�
�2
�5i
�2
�i
�7
�11
i�
7�
i�
17�
23i
�17
�i
�37
�47
i
Fin
d t
he
firs
t th
ree
iter
ates
of
the
fun
ctio
n ƒ
(z)�
z2�
c fo
r ea
chg
iven
val
ue
of c
an
d e
ach
init
ial v
alu
e.
9.c
�1
�2i
; z0
�0
10.
c�
i; z
0�
i1
�2i
�1
�i
�2
�6i
�i
�31
�22
i�
1�
i
11.
c�
1�
i; z
0�
�1
12.
c�
2�
3i; z
0�
1�
i
2�
i2
�i
4�
5i5
�7i
�8
�41
i�
22�
73i
13.B
an
kin
gM
ai d
epos
ited
$10
00 in
a s
avin
gs a
ccou
nt.
Th
ean
nu
al y
ield
on
th
e ac
cou
nt
is 5
.2%
. Fin
d th
e ba
lan
ce o
f M
ai’s
acco
un
t af
ter
each
of
the
firs
t 3
year
s.
$105
2.00
, $11
06.7
0, $
1164
.25
12-8
Answers (Lesson 12-9)
© Glencoe/McGraw-Hill A11 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill53
5A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
Co
nje
ctu
res
an
d M
ath
em
atic
al In
du
ctio
nF
requ
entl
y, t
he
patt
ern
in a
set
of
nu
mbe
rs is
not
imm
edia
tely
ev
iden
t. O
nce
you
mak
e a
con
ject
ure
abo
ut
a pa
tter
n, y
ou c
an u
sem
ath
emat
ical
indu
ctio
n t
o pr
ove
you
r co
nje
ctu
re.
1.a.
Gra
ph f
(x)�
x2an
d g
(x)�
2xon
th
e ax
es s
how
n a
t th
e ri
ght.
b.
Wri
te a
con
ject
ure
th
at c
ompa
res
n2
and
2n, w
her
e n
is a
po
siti
ve in
tege
r.If
n�
4, n
2�
2n.
c.U
se m
ath
emat
ical
indu
ctio
n t
o pr
ove
you
r re
spon
se f
rom
pa
rt b
.n
�5:
52
�25
, 25
�32
, 32
�25
Ass
ume
the
stat
emen
t is
tru
e fo
r n
�k.
Pro
ve it
is t
rue
for
n�
k�
1.(k
�1)
2�
k2
�2k
�1
�k
2�
(k�
1)k
�1
sinc
e 2
�k
�1
�k
2�
2k�
1�
ksi
nce
k2
�2k
�2k
�2k
sinc
e 1
�k
�0
�2k
�1
So
the
sta
tem
ent
is t
rue
for
n�
4.
2.R
efer
to
the
diag
ram
s at
th
e ri
ght.
a.
How
man
y do
ts w
ould
th
ere
be in
th
e fo
urt
hdi
agra
m S
4in
th
e se
quen
ce?
16 b.
Des
crib
e a
met
hod
th
at y
ou c
an u
se t
o de
term
ine
the
nu
mbe
r of
dot
s in
th
e fi
fth
dia
gram
S5
base
d on
th
e n
um
ber
of d
ots
in t
he
fou
rth
dia
gram
, S4.
Ver
ify
you
r an
swer
by
con
stru
ctin
g th
e fi
fth
dia
gram
.
Ad
d 5
to
the
num
ber
of
do
ts in
S4.
S
5w
oul
d h
ave
16 �
5 o
r 21
do
ts.
c.F
ind
a fo
rmu
la t
hat
can
be
use
d to
com
pute
th
e n
um
ber
of d
ots
in t
he
nth
dia
gram
of
this
seq
uen
ce.
Use
mat
hem
atic
al in
duct
ion
to
prov
e yo
ur
form
ula
isco
rrec
t.S n
�5
n�
4Ve
rify
that
Sn
is tr
ue fo
r n
�1.
S
1�
5(1)
�4
or 1
. Ass
ume
S nis
true
fo
r n
�k.
Pro
ve it
is tr
ue fo
r n
�k
�1.
12-9
S k�
1�
S k�
5S k
�1
�5(
k�
1)�
4�
5k�
5�
4�
(5k
�4)
�5
�S k
�5
S1
S2
S3
© G
lenc
oe/M
cGra
w-H
ill53
4A
dva
nced
Mat
hem
atic
al C
once
pts
Ma
the
ma
tic
al In
du
ctio
nU
se m
ath
emat
ical
ind
uct
ion
to
pro
ve t
hat
eac
h p
rop
osit
ion
is v
alid
for
all p
osit
ive
inte
gra
l val
ues
of
n.
1.�1 3�
��2 3�
��3 3�
�. .
.�
�n 3��
�n(n
6�1)
�
Ste
p 1
: Ver
ify t
hat
the
form
ula
is v
alid
fo
r n
�1.
Sin
ce �1 3�
is t
he f
irst
ter
m in
the
sen
tenc
e an
d�1(
16�
1)�
��1 3� ,
the
form
ula
is v
alid
fo
r n
�1.
Ste
p 2
: Ass
ume
that
the
fo
rmul
a is
val
id f
or
n�
kan
d t
hen
pro
ve t
hat
it is
als
o v
alid
fo
r n
�k
�1.
Sk
⇒ �1 3�
��2 3�
��3 3�
�. .
.�
�k 3��
�k(k 6�
1)�
Sk�
1⇒
�1 3��
�2 3��
�3 3��
. . .
��k 3�
��k
� 31
��
�k(k
6�1)
��
�k� 3
1�
� ��(k
�1 6)(k
�2)
�
Ap
ply
the
ori
gin
al f
orm
ula
for
n�
k�
1.�k
�1(
k6�
1�
1)�
or �(k
�1 6)(k
�2)
�
Thu
s, if
the
fo
rmul
a is
val
id f
or
n�
k, it
is a
lso
val
id f
or
n�
k�
1.S
ince
the
fo
rmul
a is
val
id f
or
n�
1, it
is a
lso
val
id f
or
n�
2, n
�3,
and
so o
n. T
hat
is, t
he f
orm
ula
is v
alid
fo
r al
l po
siti
ve in
teg
ral v
alue
s o
f n
.2.
5n�
3 is
div
isib
le b
y 4.
Ste
p 1
: Ver
ify t
hat
Sn
is v
alid
fo
r n
�1.
S1
⇒51
�3
�8.
Sin
ce 8
is d
ivis
ible
by
4, S
nis
val
id f
or
n�
1.S
tep
2: A
ssum
e th
at S
nis
val
id f
or
n�
kan
d t
hen
pro
ve t
hat
it is
val
id
for
n�
k�
1.S
k⇒
5k�
3�
4rfo
r so
me
inte
ger
r
Sk�
1⇒
5k�1
�3
�4t
for
som
e in
teg
er t
5k�
3�
4r5(
5k�
3)�
5(4r
)5k�
1�
15�
20r
5k�1
�3
�20
r�
125k�
1�
3�
4(5
r�
3)Le
t t
�5r
�3,
an
inte
ger
. The
n 5k�
1�
3�
4t.
Thu
s, if
Sk
is v
alid
, the
n S
k�1 is
als
o v
alid
. Sin
ce S
nis
val
id f
or
n�
1, it
is a
lso
val
id f
or
n�
2, n
�3,
and
so
on.
Hen
ce, 5
n�
3 is
div
isib
le b
y 4
for
all p
osi
tive
inte
gra
l val
ues
of
n.
k(k
�1)
�2(
k�
1)�
��
6
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
12-9
© Glencoe/McGraw-Hill A12 Advanced Mathematical Concepts
Page 537
1. D
2. C
3. A
4. B
5. A
6. A
7. A
8. B
9. C
10. D
11. D
Page 538
12. C
13. B
14. A
15. B
16. C
17. C
18. B
19. D
20. D
Bonus: C
Page 539
1. D
2. B
3. B
4. C
5. C
6. C
7. A
8. C
9. A
10. B
11. D
Page 540
12. B
13. D
14. A
15. C
16. B
17. A
18. B
19. B
20. D
Bonus: B
Chapter 12 Answer KeyForm 1A Form 1B
© Glencoe/McGraw-Hill A13 Advanced Mathematical Concepts
Chapter 12 Answer Key
Page 541
1. A
2. D
3. C
4. D
5. B
6. D
7. B
8. B
9. C
10. D
11. A
Page 542
12. B
13. A
14. D
15. A
16. C
17. B
18. C
19. A
20. A
Bonus: C
Page 543
1. �23
�
2. �7�54�
3. 537.3
4. �15
2,56625
� or 0.16384
5. 2,222,222.2
6, 6�3�, 18, 18�3�, 54 or 6, �6�3�, 18,
6. �18�3�, 54
7. does not exist
8. 12�2� � 12
9. �43925
�
10. convergent
11. divergent
Page 54427 � 9 � 3 � 1 �
�13
� � �19
�; �1892� or
12. 20�92�
13. �12
k�5 �3(2
2k
k� 1)�
1 � 5�3� � 30 �14. 30�3� � 45 � 9�3�
15. 240x3y8
16. 12.84
17. i� � 2.5416
7 � 3i, 22 � 9i,18. 67 � 27i
�3 � i, 9 � 5i,19. 57 � 89i
20. See students’ work.
Bonus: 8 � 17i
Form 1C Form 2A
© Glencoe/McGraw-Hill A14 Advanced Mathematical Concepts
Page 545
1. �4
2. �94.4
3. �1440
4. 243
5. 42.5
�4, �12, �36,�108, �324 or �4,
6. 12, �36,108, �324
7. 0
8. 36
9. �9101�
10. convergent
11. divergent
Page 546
12. 2 � 6 � 12 � 20; 40
13.
14.
15. 35,840x3y4
16. 0.7071
17. i� � 2.5953
18.
19.
20. See students’ work.
Bonus: 128
Page 547
1. 3
2. 0.3
3. 1403
4. �2560
5. �1023
6. 9, 3, 1, �13
�
7. 2
8. does not exist
9. �9593�
10. convergent
11. divergent
Page 548
12. 12 � 15 � 18 � 21; 66
13. �8
k�1�k(k
2�k
1)�
14.
15. �216xy3
16. 0.5878
17. i� + 4.0604
18.
19.
20.See students’ work.
Bonus: 32
Chapter 12 Answer KeyForm 2B Form 2C
Sample answer:
�10
n�1�n
n(n
��
11)
�
16p4 � 96p3q �
216p2q2 � 216pq3 �81q4
2 � i, 1 � 0.5i, 0.5 � 0.25i
1 � 2i, �3 � 6i,�27 � 34i
16p4 � 32p3 �24p2 � 8p � 1
2 � 8i, 4 � 16i, 8 � 32i
1 � i, 3i, �9 � i
© Glencoe/McGraw-Hill A15 Advanced Mathematical Concepts
Chapter 12 Answer KeyCHAPTER 12 SCORING RUBRIC
Level Specific Criteria
3 Superior • Shows thorough understanding of the concepts arithmetic and geometric sequences and series,common differences and ratios of terms, the binomial theorem, and mathematical induction.
• Uses appropriate strategies to solve problems and prove a formula by mathematical induction.
• Computations are correct.• Written explanations are exemplary.• Word problems concerning arithmetic and geometric sequences are appropriate and make sense.
• Goes beyond requirements of some or all problems.
2 Satisfactory, • Shows understanding of the concepts arithmetic and with Minor geometric sequences and series, common differences Flaws and ratios of terms, the binomial theorem, and
mathematical induction.• Uses appropriate strategies to solve problems and prove a formula by mathematical induction.
• Computations are mostly correct.• Written explanations are effective.• Word problems concerning arithmetic and geometric sequences are appropriate and make sense.
• Satisfies all requirements of problems.
1 Nearly • Shows understanding of most of the concepts arithmetic Satisfactory, and geometric sequences and series, common differences with Serious and ratios of terms, the binomial theorem, and Flaws mathematical induction.
• May not use appropriate strategies to solve problems or prove a formula by mathematical induction.
• Computations are mostly correct.• Written explanations are satisfactory.• Word problems concerning arithmetic and geometric sequences are appropriate and sensible.
• Satisfies most requirements of problems.
0 Unsatisfactory • Shows little or no understanding of the concepts arithmetic and geometric sequences and series, common differences and ratios of terms, the binomial theorem, and mathematical induction.
• May not use appropriate strategies to solve problems or prove a formula by mathematical induction.
• Computations are incorrect.• Written explanations are not satisfactory.• Word problems concerning arithmetic and geometric sequences are not appropriate or sensible.
• Does not satisfy requirements of problems.
© Glencoe/McGraw-Hill A16 Advanced Mathematical Concepts
Chapter 12 Answer Key
Page 5491a. Sample answer: Mr. Ling opened a savings
account by depositing $50. He plans todeposit $25 more per month into theaccount. What is his total deposit afterthree months? The sequence is 50 �(n � 1)25, and $100 is his total deposit afterthree months.
1b. Sample answer: The common difference is$25. The nth term is $50 � (n � 1)$25.
1c. Sample answer:
S12 � �122�(50 � 325) � 2250
S12 � 50 � 75 � 100 � 125 � 150 � 175 �
200 � 225 � 250 � 275 � 300 � 325
S12 � (50 � 325) � (75 � 300) �(100 � 275) � (125 � 250) �(150 � 225) � (175 � 200)
Since the sums in parentheses are all equal.
S12 � 6(50 � 325), or �122� (50 � 325), or
�2n� (a1 � an)
1d. No; arithmetic series have no limits; it isdivergent.
2a. Sample answer: Mimi has $60 to spend onvacation. If she spends half of her moneyeach day, how much will she have left afterthe third day?
$60 � ��12
��3
� $7.50
After the third day, she has $7.50.
2b. Sample answer: The common ratio is �12
�.The nth term is 60��1
2��
n�1.
2c. Sample answer:
S11 � � 120
2d. If r � limn→∞
�aan�
n
1� � 1, the series
converges; r � �12
�; the series converges.
3a. Prove that the statement is true for n � 1.Then prove that if the statement is true forn, then it is true for n � 1.
3b. Here Sn is defined as
a1 � a1r � a1r2 � . . . � a1r
n�1 � �a1
1�
�
ar1r
n
�
Step 1: Verify that the formula is valid for n � 1.
Since S1 � a1 and S1 � �a1
1�
�
ar1r
1
�
� �a1
1(1
�
�
rr)
�
� a1,
the formula is valid for n � 1.
Step 2: Assume that the formula is for n � k and derive a formula for n � k � 1.
Sk ⇒ a1 � a1r � a1r2 � . . . � a1r
k�1 �
�a1
1�
�
ar1r
k
�
Sk�1 ⇒ a1 � a1r � a1r2 � . . . � a1r
k�1 �
a1rk�1 � 1
� �a1
1�
�
ar1r
k
� � a1rk�1 � 1
� �a1
1�
�
ar1r
k
� � a1rk
�
�
Apply the original formula for n � k � 1.
Sk�1 ⇒�a1 �
1a�
1rr
( k�1)
�
The formula gives the same result as adding the (k � 1) term directly. thus, if the formula is valid for n � k, it is also validfor n � k � 1. Since the formula is valid forn � 2, it is valid for n � 3, it is also valid forn � 4, and so on indefinitely. Thus, theformula is valid for all integral values of n.
4. From the binomial expansion, the fourth term
of ���y2x�� � �
�yx�
��6is
�36!3!!
����y2x���3���
�yx�
��3� �20 �
y1
3� or ��2y03�.
a1 � a1rk�1
��1 � r
a1 � a1rk � a1r
k � a1rk�1
����1 � r
60 � 60��12
��11
��1 � �
12�
Open-Ended Assessment
© Glencoe/McGraw-Hill A17 Advanced Mathematical Concepts
Mid-Chapter TestPage 550
1. 129
2. 1175
3. 27 or 128
4. �29,524
5. 64, �32, 16, �8
6. �21�
7. does not exist
8. �6939� or �
171�
9. convergent
10. divergent
Quiz APage 551
1. �9�3� � 9�5�
2. 17
3. 103.7
4. 9�5�
5. 20.78125
6. �13
�, � ��95��, �
257�, . . .
Quiz BPage 551
1. does not exist
2. 1
3. 0
4. �31�
5. does not exist
6. �4959� or �
151�
7. divergent
8. divergent
9. convergent
10. divergent
Quiz CPage 552
1. 2�12
� � 4�12
� � 8�12
�; 15�12
�
2. ��
n�1�1861� ��
23��
n�1
3. �100
k=1(2k � 1)(2k)
81a4 � 108a3d �4. 54a2d2 � 12ad3 � d4
5. 36.77
6. 0.8660
Quiz DPage 552
1. 1.1, 1.11, 1.111, 1.1111
2. 6 � i, 12 � i, 24 � i
3. �2 � 2 i, �1 � 6i, �36 �14i
4. See students’ work.
5. See students’ work.
Chapter 12 Answer Key
Sample answer:
© Glencoe/McGraw-Hill A18 Advanced Mathematical Concepts
Page 553
1. D
2. A
3. C
4. C
5. D
6. E
7. E
8. A
9. C
Page 554
10. D
11. D
12. D
13. E
14. B
15. D
16. B
17. A
18. D
19. 4000
20. 48
Page 555
1. (�5, 7)
No, ƒ(x) is2. undefined when x � 3.
3. (x � 2) (2x � 1) (x � 3)
4. y � �sin �3x � ��5
�� � 2
5. �152�92�9��
6. 560.2 N, 32.2
7. 12�cos �96�� � i sin �96
���; �12i
15(x�)2 � 30 x�y� �8. 15( y�)2 � 108
9. $1704
10. �10
n�1�(�
21n)n
�
�1
13n
�
Chapter 12 Answer KeySAT/ACT Practice Cumulative Review
© Glencoe/McGraw-Hill A19 Advanced Mathematical Concepts
Trigonometry Semester Test
Page 557
1. B
2. C
3. B
4. D
5. D
6. B
7. C
8. B
9. A
Page 558
10. C
11. D
12. C
13. D
14. C
15. B
16. C
17. D
18. C
19. C
Page 559
20. A
21. B
22. A
23. B
24. A
25. D
Answer Key
© Glencoe/McGraw-Hill A20 Advanced Mathematical Concepts
Page 560
26. ��2, 14, 10�
27. y � ��21�x � �1
21�
28. 0.79
29. 1, 3, 9, 27
30. �2�2�, ��4
��
31. (�1, 1)
32. cos 120
33. 0, 30
34. �4�13
1�3��
35. ��47��
36. �5
k�12k
37. 3
38. 4.3938
Page 56139. 3
40. �48,384c5d3
41. �4, 4, 2�
42. �3�13
1�3��
43. �9, 11�
r � csc � or44. 1 � r cos �� � ��
2��
45. x � 0
46. �1�1�
47. yes
48. 11.64
49. �171�
50.
TrigonometryFinal TestPage 563
1. D
2. C
3. C
4. B
5. C
Answer Key
© Glencoe/McGraw-Hill A21 Advanced Mathematical Concepts
Page 5646. A
7. A
8. D
9. C
10. D
11. B
12. A
13. C
14. B
Page 565
15. A
16. D
17. B
18. C
19. B
20. D
21. D
22. B
23. C
Page 566
24. B
25. B
26. B
27. A
28. C
29. D
30. B
Answer Key
© Glencoe/McGraw-Hill A22 Advanced Mathematical Concepts
Page 567
31. �4
32. 618.5 cm2
33. � �34. (�1, 3)
35. y � �3x�� 1
36.
37.
38. �1475�
39. � �40. 1
41. 8.1 ft
42. 1541
Page 568
43.
44. 19 yrs
45.
46. 1.3917
47. �1.3673
48. 1,594,322
49. ��6� �4
�2��
50. � �
51. 11
Page 569
52. 2.52 cm
53.
54. log3 �19
� � �2
55. ��6� �4
�2��
56. 0.62
57. 3i� � 4j� � 2k�
58. �9
n�1(2n � 1)
59. �1�3� (cos 5.30 � i sin 5.30)
60. 15 � 8i
61. (�2, �2)
62. �4; imaginary
63. 60°
64. 5; 3
�25
1�2
02
113
27
30
Answer Key
yesD � {�2, 0, 4, 9}R � {�3, 3, 5}
Sample answer: (2, 135°), (�2, �45°),(2, �225°), (�2, 315°)
© Glencoe/McGraw-Hill A23 Advanced Mathematical Concepts
Page 570
65. y � �13
�x � �23
�
66. (2, 0, �1)
67. 5
68. �21�
69. 36
70. �32�
71. 3 or 1
72. ��13
�, ��32�, �1, �2
73. 28.9 cm
74.
75.
Answer Key
c � 66.4, C � 76.7°,b � 66.3
a � 6.9, B � 107.9°,C � 32.1°
BLANK