Algebra 2: Chapter 12 Notes Packet Name:______________________________

**Included inside: Assignment Sheet, Notes, Study Guide, Homework Worksheets**

Algebra 2 Chapter 12 Assignment Sheet*All assignments are due the day after they are assigned*

Date In-Class Homework

33W

1/1212.2: Arithmetic Sequences

HW33:Pg. 806 #4-20evenHW#33 Worksheet (pg 21 in this packet)

34Th

1/1312.3: Geometric Sequences and Series

HW34:Pg. 814 #4-20evenHW#34 Worksheet (pg 20 in this packet)

35F

1/14

12.4: Sums of Infinite Geometric Sequences

HW35:Pg. 823 #8-22 evenHW#35 Worksheet (pg 19 in this packet)

**No School on Monday 1/17**

36T

1/1812.1: Sums of Arithmetic Sequences

HW36:Pg. 798 #2-26 even, 27, 38-58 evenHW#36 Worksheet (pg 18 in this packet)Chapter 12 Test and Notes Check on Thursday

37W

1/19Chapter 12 Review

HW37:Chapter 12 review sheet (pgs. 15-17 in this packet)

Chapter 12 Test and Notes Check on Thursday

38Th

1/20Chapter 12 Test and Notes Check

HW38:Chapter 7 final review sheet Chapter 8 final review sheetPrint: Final Review Packet

Practice Final on Tuesday, 1/25Final Exam for periods 1, 2 on Thursday, 1/27 for periods 3, 4 on Friday, 1/28

1 of 21

Chapter 12 Sequences and Series

Notes 33 : Section 12-2: Arithmetic Sequences

Definition of an Arithmetic SequenceAn arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference, d, of the sequence.

Formula for the General Term ( ) of an Arithmetic SequenceThe nth term (the general term) of an arithmetic sequence with first term a1 and common difference d is:

Practice: In 1-4, write the formula for the general term of the arithmetic sequence then find the indicated term using the given information.

1. Find and when = 9 and d = 2. 2. Find and when d = −3 and = 73

3. Find and when = 34 and = 122 4. Find and when = 8 and = -136

2 of 21

The Sum of the First n Terms of an Arithmetic SequenceThe sum, , of the first n terms of an arithmetic sequence is given by

In which is the first term and is the nth term.

For Examples 5-7, find each sum.5.) Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,…..

6.) Find the sum of the first 60 positive event integers.

7.) Find the sum of the even integers between 21 and 45.

Sigma Notation:

8.) Evaluate:

9.) Evaluate:

3 of 21

10.) Find n given = 175 for the arithmetic sequence: 4, 7, 10, 13, 16, 19,…….

11.) Find n given = 366 for the arithmetic sequence: 3, 8, 13, 18, 23,……

12.) Find n given = 308 for the arithmetic sequence: 3, 8, 13, 18, 23,……

4 of 21

Exponential and Logarithmic Equation Review*You must do these problems in order to get full credit for your notes, even if we do not do them in class.

For #13-18, solve the exponential equation. Give the approximate value rounded to three decimal places if necessary.13. 14. 15.

16. 17. 18.

In #19-24, Solve the logarithmic equation. Give the approximate value rounded to three decimal places if necessary.19. 20. 21.

22. 23. 24.

HW #33: Pg. 806 #4-20even; HW#33 Worksheet (pg 21 in this packet)

5 of 21

Notes 34: Section 12-3: Geometric Sequences and Series

Definition of a Geometric SequenceA geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.

Formula for the General Term ( ) of a Geometric SequenceThe nth term (the general term) of a geometric sequence with first term and common ratio r is:

1.) Write the first five terms of each geometric sequence given: = 5 and r = 3.

For #2-4, write the rule for the nth term of the geometric sequence using the given information.

2. = 4 and r = 2 3. = 45 and = −1215

4. = 18 and =

6 of 21

The Sum of the First n Terms of a Geometric SequenceThe sum, , of the first n terms of a geometric sequence is given by

In which is the first term and r is the common ratio (r ≠ 1).

Examples: For #5-8, find the indicated sum for the geometric sequence.

5. Find the sum of the first 10 terms of: 1, 5, 25, 125, 625,……. 6.

7. 8.)

7 of 21

Examples: For #9-10, find the number of terms n need to get the given sum .

9. = 3906 given 1, 5, 25, 125, 625,…… 10. given 4, 2, 1, ……

Exponential and Logarithmic Equation Review*You must do these problems in order to get full credit for your notes, even if we do not do them in class.For #11-16, solve the exponential equation. Give the approximate value rounded to three decimal places if necessary.

11. 12. 13.

14. 15. 16.

HW #34: Pg. 814 #4-20even; HW#34 Worksheet (pg 20 in this packet)

8 of 21

Notes #35: Section 12-4: Finding Sums of Infinite Geometric Series

The Sum of an Infinite Geometric SeriesThe sum of an infinite geometric series with first term and common ratio r is given by

provided . If , the series has no infinite sum.

Examples: For #1-4, find the sum of each infinite geometric series.

1.) 2.)

3.) 4 + 8 + 16 + …. 4.)

5.) Find the common ratio of an infinite

geometric series with first term = and has

a sum of −2.

6.) Find the common ratio of an infinite geometric series with first term = 28 and has a sum of 35.

9 of 21

Sequence and Series Review:7.) Find the 25th term of the sequence: 1, -2, -5, -8, …

8.) Find the 16th term of the sequence: -108, 36, -12, 4, …

9.) Given an arithmetic sequence, if and , find

10.) Evaluate:

11.) Given the arithmetic sequence such that and , find

12.) Given an arithmetic sequence such that = 7 and = 205, find and .

13.) Given the geometric sequence such that

and , find

14.) Given an arithmetic sequence such that and d = 3, find n given

10 of 21

Exponential and Logarithmic Equation Review

15.) Graph . Graph the parent function then transform the graph using h and k values. Then state the domain, range and the equation of asymptote for the function.

16.) Write with the common base. 17.) Solve:

For #18-19, use and to approximate the value of each expression.18.) 19.)

HW #35: Pg. 823 #8-22 even; HW#35 Worksheet (pg 19 in this packet)

11 of 21

Notes #36: Section 12-1: Sequences and Summation NotationAfter completing section 12-1 you should be able to do the following:

1. Find the terms of a sequence given the rule.2. Write the rule for a given sequence.3. Use summation notation.4. Memorize the formulas to find sums of special series and use them to find the sum

Definition of a Sequence

A sequence is a function whose domain is a set of consecutive integers. If the domain is not specified, it is understood that the domain starts with 1. The values in the range are called terms of the sequence.

Domain: 1 2 3 4……. n The relative position of each term

Range: .… Terms of the sequence

A finite sequence has a limited number of terms. An infinite sequence continues without stopping.

Finite Sequence: 2, 4, 6, 8 Infinite Sequence: 2, 4, 6, 8, ……

A sequence can be specified by an equation, or rule. For example both sequences above can be described by the rule:

= 2n

Examples: For #1-2, write the first four terms of each sequence whose rule is given.

1. 2.

Finding the rule of a sequence given its terms.In order to write a rule you have to pay attention to its domain also know as the term number, n. The term number is its position in the sequence. The symbol used to show the value of the term is , so the first term is the second term is and so on. In order to write the rule it is all about describing the patterns you see between the term number and the value of the term. It is really about guess and check at this point. So let’s try some.

12 of 21

Examples: For #3-4, write the rule, , for the given sequence.3.) 1, 6, 11, 16,….. 4.) 1.2, 4.2, 9.2, 16.2,…..

5.) 6.) –4, 8, –12, 16,……

Summation NotationIt is sometimes useful to find the sum of the first n terms of a sequence. A compact notation of expressing the sum of the first n terms is called summation notation.

Summation NotationThe sum of the first n terms of a sequence ins represented by the summation notation

where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Examples: For #7-8, expand and evaluate the sum.

7. 8.

13 of 21

Writing summation notation

In order to write a series in summation notation you must find the rule ( ). Once you have found the rule that relates the term number to its value need to figure out the number of terms in the sequence given. If it goes on without end we state the limit of summation is ∞.

Examples: For #9-10, write the series using summation notation.

9.) 7 + 10 + 13 + 16 + 19 10.)

Special Formulas to find sums of series without having to add up each individual term of the series:Sum of n terms of a constant c:

Sum of first n positive integers:

Sum of squares of first n positive integers:

Only use these formulas when n (upper limit of summation is a large number)

Examples: For #11-13, find the sum of the series.

11.) 12.) 13.)

HW #36: Pg. 798 #2-26 even, 27, 38-58 even; HW#36 Worksheet (pg 18 in this packet)

14 of 21

HW #37--Chapter 12 Review Sheet Name: _____________________________

For question 1-3, write a rule for the nth term of the arithmetic sequence. Then find .

1.) 1, 7, 13, 19, 25, … 2.) ,… 3.) -4, -2, 0, 2, 4,…

For questions 4-6, write a rule for the nth term of the arithmetic sequence.4.) d = -4, =7 5.) 6.)

For questions 7-8, find (a) the sum of the first n terms of the arithmetic series and (b) find n for the given sum, .

7.) (a) n = 50 (b) =1005

8.) 32, 24, 16, 8, 0, … (a) n = 30 (b) = -880

For questions 9-10, write a rule for the nth term of the geometric sequence.

9.) 10.)

For questions 11-12, find (a) the sum of the first n terms of the geometric series and (b) find n for the given sum, .11.) (a) n = 8 (b) = 819

12.) (a) n = 20 (b) =5.5

(more on next page!!)15 of 21

For questions 13-15, find the sum of the series.

13.) 14.) 15.)

For questions 16-18, find the sum of the infinite geometric series if it has one.

16.) 17.) 18.)

For questions 19-20, find the common ratio of the infinite geometric series with the given sum and first term.

19.) 20.)

For questions 21-22, find the first five terms of each sequence.

21.) 22.)

In 23-25, evaluate.

23.) 24.) 25.)

(more on next page!!)

16 of 21

In 26-34, solve each equation. (a) Give the exact value (before plugging anything into your calculator). (b) Round to three decimal places, if needed.

26.) 27.) 28.)

29.) 30.) 31.)

32.) 33.) 34.)

(now you’re done! )

17 of 21

HW#36 Worksheet Name: ______________________For questions #1-3, write a rule for the nth term of the arithmetic sequence. Then find .

1.) 3, 7, 11, 15, 19, … 2.) ,… 3.) -3, -1, 1, 3, 5,…

For questions #4-6, write a rule for the nth term of the arithmetic sequence.4.) d = -6, =7 5.) 6.)

For questions #7-8, find (a) the sum of the first n terms of the arithmetic series and (b) find n for the given sum, .

7.) (a) n = 40 (b) =135

8.) 32, 24, 16, 8, 0, … (a) n = 50 (b) = -5544

In #9-17, solve the final exam review problems. Leave your answers in (a) exact form and (b) in approximate form, rounded to four decimal places.9. 10. 11.

12. 13. 14.

15. 16. 17.

18 of 21

HW#35 Worksheet Name: ______________________For #1-8, complete each problem about sequences and series. 1. Write a rule for the arithmetic sequence: -3, 5, 13, 21, …

2. Write a rule for the geometric sequence:

3. Find n for the sequence 4, -3, -10, -17, … if 4. Find n for the sequence -2, 6, -18, 54, … if

5. Find and for the sequence: 97, 75, 53, 31…

6. Find of the geometric sequence for whom

and

7. Evaluate: 8. Evaluate:

In #9-17, solve the final exam review problems. Leave your answers in (a) exact form and (b) in approximate form, rounded to four decimal places.9. 10. 11.

12. 13. 14.

15. 16. 17.

19 of 21

HW#34 Worksheet Geometric Sequences Name: _____________________Formulas you need to memorize for Geometric Sequences:

General Rule: Sum:

For 1-4, find the value of ‘n’ for the given sum for each problem.1. 2.

3. 4.

Find the rule for the geometric sequence describe by the given information.5. a4 = 108; r = 3 6. a5 = 20; a9 = 320 7. a2 = 4, a4 = 5 8. a2 = -24, a5 = 1536

In #9-17, solve the final exam review problems. Leave your answers in (a) exact form and (b) in approximate form, rounded to four decimal places.9. 10. 11.

12. 13. 14.

15. 16. 17.

20 of 21

HW#33 Worksheet: Arithmetic Sequences Name: ____________________Formulas you need to memorize for Arithmetic Sequences:

General Rule: Sum:

In #1-6, Write a rule for the nth term of the arithmetic sequence with the given information.

1. d = 14, a14 = 46 2. a2 = −28, a20 = = 52 3. a1 = −2, a9 =

4. a5 = 17, a17 = 77 5. d = , a8 = 24 6. d = 4.1, a16 = 48.2

In #7-10, Find n for the given sum Sn.7. 3 + 8 + 13 + 18 + 23 +……; Sn = 366 8. 50 + 42 + 34 + 26 + 18 +…..; Sn = 182

9. 2 + 9 + 16 + 23 + 30 + …..; Sn = 1661 10. 2 + 16 + 30 + 44 + 58 + ….; Sn = 2178

In #11-22, solve the final exam review problems. Leave your answers in (a) exact form and (b) in approximate form, rounded to four decimal places.11. 12. 13.

14. 15. 16.

17. 18. 19.

20. 21. 22.

21 of 21