Chapter 10 Notes: Sequences, Induction, and Probabilityteachers.sduhsd.net/chayden/documents/Chapter10Note... · Web viewA finite sequence has a limited number of terms. An infinite

Chapter 10 Notes: Sequences, Induction, and ProbabilityDay 31: Section 10-1: Sequences and Summation Notation

After completing section 10-1 you should be able to do the following: 1. Find particular terms of a sequence from the general term.2. Use recursion formulas.3. Use factorial notation.4. Use summation notation.

Finding the term of a sequence if given a rule for the term is as simple as evaluating the rule for the term number.

Practice: 1-2, Write the first four terms of each sequence whose rule is given.

1. 2.

Recursion Formulas

A recursion formula defines the nth term of a sequence as a function of the previous term. When you use a recursion formula you can only get the next term of the sequence and you must be given the prior term.

Example of using a recursion formula: Write the first four terms of the sequence that are defined by using the recursion formula of: a1 = 7 and an = 2an−1 − 5 for n ≥ 2.

Math Analysis Notes prepared by Mrs. Atkinson 1

Definition of a SequenceA 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: a1 a2 a3 a4……. an 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: an = 2n

Practice: 1-2, are defined using recursion formulas. Write the first four terms of each sequence.1. a1 = 4 and an = an−1 + 3 for n ≥ 2. 2. a1 = 0 and an = −an−1 + 3

Factorial NotationProducts of consecutive positive integers occur quite often in sequences. These products can be expressed in a special notation, called factorial notation.

Examples of values of n! for the first 5 positive integers are:1! = 12! = 2∙1 = 23! = 3∙2∙1 = 64! = 4∙3∙2∙1 = 245! = 5∙4∙3∙2∙1 = 120

Practice: In 1-2, evaluate the expression.1. 3∙4! 2. (3∙4)!

Practice: In 1-2, write the first four terms of the sequence whose nth term given:

1. 2.


Factorial NotationIf n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n down through 1.

n! = n(n – 1)(n – 2)……(3)(2)(1)0! (zero factorial), by definition is 1.

0! = 1

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.

Practice: In 1-2, Expand and evaluate the sum.

1. 2.

Practice: In 1-2, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.

1. 4. 1 + 3 + 5 +….+(2n – 1)


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.

Asn 31: p933#3-54 by 3’s, 62-68(even) Chapter P Final Exam Review Worksheet #2-23 by 3’s

Day 33: Section 10-2: Arithmetic Sequences

After completing section 10-2 you should be able to do the following: 1. Find the common difference for an arithmetic sequence.2. Write terms of an arithmetic sequence.3. Use the formula for the general term of an arithmetic sequence.4. Use the formula for the sum of the first n terms of an arithmetic sequence.

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 an and a16 when a1 = 9 and d = 2. 2. Find an and a7 when d = −4.1 and a16 = 48.2

3. Find an and a5 when a7 = 34 and a18 = 122 4. Find an and a6 when a2 = −28 and a20 = 52

Practice: Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,…..


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 of the sequence.

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

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

In which a1 is the first term and an is the nth term.

Practice: Find the sum of the first 60 positive event integers. Practice: Find the sum of the even integers between 21 and 45.

Practice: Evaluate:

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

Practice: Find n given Sn = 366 for the arithmetic sequence: 3, 8, 13, 18, 23,……


Asn 33: p943#3-48 by 3’s Arithmetic Worksheet (all)

Chapter P Final Exam Worksheet #3-24 by 3’s

Day 34: Section 10-3: Geometric Sequences and Series

After completing section 10-3 you should be able to do the following: 1. Find the common ratio of a geometric sequence.2. Write terms of a geometric sequence.3. Use the formula for the general term of a geometric sequence.4. Use the formula for the sum of the first n terms of a geometric sequence.5. Use the formula for the sum of an infinite geometric series.

Practice: Write the first five terms of each geometric sequence given: a1 = 5 and r = 3.

In 1-3, Write the rule for the nth term of the geometric sequence using the given information.1. a1 = 4 and r = 2 2. a2 = 45 and a5 = −1215

3. a3 = 18 and a6 =


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 (an) of a Geometric SequenceThe nth term (the general term) of a geometric sequence with first term a1 and common ratio r is:

Practice: In 1-2, Find the indicated sum for the geometric sequence.

1. Find the sum of the first 10 terms of: 1, 5, 25, 125, 625,……. 2.

Practice: In 1-2, Find the number of terms n need to get the given sum Sn.

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

Practice: In 1-2, Find the sum of each infinite geometric series.

1. 2.


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

In which a1 is the first term and r is he common ratio (r ≠ 1).

The Sum of an Infinite Geometric SeriesThe sum of an infinite geometric series with first term a1 and common ratio r is given by

provided . If , the series has no infinite sum.

Practice: Find the common ratio of an infinite geometric series with first term a1 = and has a sum of −2.

Day 35: Section 10-4: Mathematical Induction

After completing section 10-4 you should be able to do the following: 1. Write the statement for Sk and Sk + 1.2. Use mathematical induction to prove statements about all positive integers.

Mathematical Induction works as follows. If you know from the basis step that a statement is true for n = 1, then the inductive step implies that it is true for n = 2, and therefore, n = 3, and so on for all positive integers n.

Example Use mathematical induction to prove the statement.

4 + 7 + 10 + 13 + …..+ (3n + 1)=


Asn 34: p955#3-42 by 3’s Geometric Sequence Worksheet

Chapter 1 Final Exam Worksheet #1-16 by 3’s

Mathematical InductionTo show that a statement is true for all positive integers n, perform these steps:

1. Show that the statement is true for n = 1.2. Assume that the statement is true for n = k.3. Prove that the statement is true for n = k + 1.

Practice: A statement for Sn about the positive integers is given. Write statements S1, S2, and S3 and show that each of these statements is true.

Sn: 3 + 4 + 5 +……+ (n + 2) =

Practice: A statement for Sn about the positive integers is given. Write statements Sk, and Sk+1, make sure to simplify statement Sk+1 completely.

Sn: 4 + 8 + 12 +…..+ 4n = 2n(n + 1)

Practice: Use mathematical induction to prove that each statement is true for every positive integer n.

1 + 2 + 22 +……+ 2n − 1 = 2n – 1


Asn 35: p965#2-18(even) Chapter 1 Final Exam Worksheet #2-17 by 3’s Study for Chapter 10 Quiz Sections 1-3

Day 36: Quiz Chapter 5 Sections 1-3

Day 37: Section 10-5: The Binomial Theorem

After completing section 10-5 you should be able to do the following: 1. Expand a binomial raised to a power using the Binomial Theorem

Pascal’s Triangle:Pascal's Triangle is an arithmetical triangle you can use for some neat things in mathematics. Here's how you construct it: 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 1 7 21 35 35 21 7 1 . . .You start out with the top two rows: 1, and 1 1. Then to construct each entry in the next row, you look at the two entries above it (i.e. the one above it and to the right, and the one above it and to the left). At the beginning and the end of each row, when there's only one number above, put a 1. You might even think of this rule (for placing the 1's) as included in the first rule: for instance, to get the first 1 in any line, you add up the number above and to the left (since there is no number there, pretend it's zero) and the number above and to the right (1), and get a sum of 1.

Practice: What is the 8th row of Pascal’s Triangle.

Practice: In 1-2, Use the Binomial Theorem to expand each binomial.

1. (2x – 3y)4 2. (x + 2y)5


The Binomial TheoremThe binomial expansion of (a + b)n for any positive integer n is:

(a + b)n = nC0anb0 + nC1an – 1b1 + nC2an – 2b2 +…….+ nCna0bn

The red parts of the binomial expansion are called the binomial coefficients.You can find the binomial coefficients by using Pascal’s Triangle.

Asn 36: p966#20-24(even) Chapter 1 Final Exam Worksheet #3-15 by 3’s

Asn 37: p972#10-30(even) Chapter 2 Final Exam Worksheet #1, 5, 10

Day 38: Section 10-5: The Binomial Theorem(continued) Section 10-6: Counting Principles, Permutations, and Combinations

After completing section 10-5 and 10-6 you should be able to do the following: 1. Find a particular term in a binomial expansion.2. Use the Fundamental Counting Principle.3. Use the permutations formula.4. Use the combinations formula.5. Distinguish between permutation problems and combination problems.

Finding a Particular Term in a Binomial ExpansionBy observing the terms in the formula for expanding binomials, we can find a formula for finding a particular term without writing the entire expansion:

(a + b)n = nC0anb0 + nC1an – 1b1 + nC2an – 2b2 +…….+ nCna0bn

Use the following formula to find a particular term in a Binomial ExpansionThe (r + 1)st term of the expansion of (a + b)n is:

nCran – rbr

Practice: In 1-2, find the indicated term of the binomial expansion.1. Find the 3rd term of (x – 2y)5 2. Find the sixth term of (2x + 3y)9

The Fundamental Counting PrincipleThe number of ways in which a series of successive things can occur is found by multiplying the number of ways each thing can occur.

Example of Fundamental Counting Principle:A pizza can be ordered with three choices of size (small, medium, or large), four choices of crust (thin, think, crispy, or regular), and six choices of toppings(ground beef, sausage, pepperoni, bacon, mushrooms, or onions). How many different one-topping pizzas can be ordered?

Event #1: Size Event #2: Crust Event #3: Toppings


First Term

SecondTerm Third

Termnth Term

The exponent of b is 1 less than the term number

Total number of one-topping pizzas that can be ordered: 3∙4∙6 = 72

3 4 6Practice: You are taking a multiple-choice test that has six questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each questions and leave nothing blank, in homw many ways can you answer the questions?

Practice: License plates in a state display two letters followed by three digits. How many different license plates can be manufactured (a) if you are allowed to repeat letters and digits and (b) if you are not allowed to repeat letters and digits.

PermutationsA permutation is an ordered arrangement of items.The number of possible permutations if r items are taken from n items is

Practice: A corporation has seven members on its board of directors. In how many different ways can it elect a president, vice-president, secretary, and treasurer?

Practice: In how many ways can 6 different books be lined up?

CombinationsOrder does not matter.The number of possible combinations if r items are taken form n items is

Practice: From a group of 10 doctors, in how many was can four of them be selected to attend a conference?


Practice: How many different 5-card poker hands are possible from a standard 52-card deck?

Day 39: Section 10-7: ProbabilityAfter completing section 10-7 you should be able to do the following:

1. Compute theoretical probability.2. Find the probability of one event or a second event occurring.3. Find the probability of one event and a second event occurring.

Theoretical ProbabilityIf an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the theoretical probability of event (E), denoted by P(E), is:

Example: A die is rolled. Find the probability of getting a number less than 5.

Solution: n(E) = 4 since the numbers on a die less than 5 are: 1, 2, 3, and 4; n(S) = 6 since there are six sides of the dice each one with a different number so:

Practice: What is the probability of getting a sum of 5 when two six-sided dice are rolled?

Practice: If you are dealt one card from a standard 52-card deck, find the probability of being dealt a king.


Asn 38: p972#40-46(even) p981#3-66 by 3’s Chapter 2 Final Exam Worksheet #2, 6, 11

Practice: What is the probability of getting a 5-card hand that contains two aces and three other cards (not aces) from a standard deck of 52 cards.

Practice: What is the probability of getting a 5-card hand that contains 3-cards of the same suit and 2-cards of a different suit.

Or Probabilities with Mutually Exclusive Events (events that have nothing in common).If A and B are mutually exclusive events, then

P(A or B) = P(A) + P(B)

Practice: If one card is randomly selected from a deck of 52 cards, what is the probability of selecting a king or a seven?

Practice: If you roll a single, six-sided die, what is the probability of getting either a 4 or a 5?

Or Probabilities with Events That are Not Mutually Exclusive (events that have something in common).If A and B are not mutually exclusive events, then

P(A or B) = P(A) + P(B) – P(A and B)where P(A and B) is the overlap when both events happen at the same time.

Practice: If one card is randomly selected from a deck of 52 cards, what is the probability of selecting a king or a heart?


Practice: If you roll a single, six-sided die, what is the probability of getting either a 4 or an even number?

And Probabilities with Independent EventsIf A and B are independent events, then

P(A and B) = P(A)∙P(B)Practice: Find the probability of a family having four boys in a row.

Practice: Find the probability a single die that is rolled twice will have an even number the first time it is rolled and a number greater than 2 the second time rolled.

Chapter 10 Review Sheet

Complete each problem on a separate sheet of paper. Show all of your work! NO WORK = NO CREDIT!

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 the sum of the first n terms of the arithmetic series and

7. , n = 50 8. 32, 24, 16, 8, 0, …, n = 30


Asn 39: p994#1-26(all), 29, 30, 37-42(all), 47-52(all) Chapter 2 Final Exam Worksheet #3, 4, 7, 8, 9

For questions 9-10, find n for the given sum, .

9. , =1005 10. 32, 24, 16, 8, 0, …, = -880

For questions 11-12, write a rule for the nth term of the geometric sequence.

11. 12.

For questions 13-14, find the sum of the first n terms of the geometric series and

13. , n = 8 14. , n = 20

For questions 15-16, find n for the given sum, .

15. , =-819 16. , =5.5

For questions 17-19, find the sum of the series.

17. 18. 19.

For questions 20-22, find the sum of the infinite geometric series if it has one.

20. 21. 22.

For questions 23-28, solve each problem.

23. Tim wants to buy one of three different albums he sees in a music store. Each is available on tape and on CD. From how many combinations of album and format does he have to choose?

24. A video store has 8 new releases this week. Each is available on videotape and on DVD. How many ways can a customer choose a new release and a format to rent?

25. Carlos has homework to do in math, chemistry, and English. How many ways can he choose the order in which to do his homework?


26. The menu for a banquet has a choice of two types of salad, five main courses, and three desserts. How many ways can a salad, main course, and dessert be selected to form a meal?

27. How many ways can six different books be arranged on a shelf if one of the books is a dictionary and it must be on an end?

28. Each question on a five-question multiple-choice quiz has answer choices labeled A, B, C, and D. How many different ways can a student answer the five questions?

For questions 29-29, Find the number of possibilities.

29. The winner and first, second, and third runners-up in a contest with 10 finalists.

30. Selecting two of eight employees to attend a business seminar.

31. An arrangement of the letters in the word “algebra”

32. Placing an algebra book, a geometry book, a chemistry book, an English book, and a health book on a shelf.

33. Selecting nine books to check out of the library from a reading list of twelve.

34. Choosing two CDs to buy from ten that are on sale.

35. Selecting three of fifteen flavors of ice cream at the grocery store.

36. Choosing a hand of five cards from a standard deck of cards consisting of four cards from one suit and one card from another suit.

37. An airline is hiring 5 flight attendants. If 8 people apply for the job, how many different groups of 5 attendants can the airline hire?

38. A photographer is taking pictures of a bride and groom and their 6 attendants. If she takes photographs of 3 people in a group, how many different groups can she photograph?

39. A school librarian would like to buy subscriptions to 7 new magazines. His budget, however, will allow him to buy only 4 new subscriptions. How many different groups of 4 magazines can he choose?


For questions 40-43, Lisa has 9 rings in her jewelry box. Five are gold and 4 are silver. If she randomly selects 3 rings to wear to a party, find each probability.40. P(2 silver or 2 gold) 41. P(all gold or all silver)42. P(at least 2 gold) 43. P(at least 1 silver)

For questions 44-49, seven girls and six boys walk into a video store at the same time. There are five salespeople available to help them. Find the probability that the salespeople will first help the given number of girls and boys.44. P(4 girls or 4 boys) 45. P(3 girls or 3 boys) 46. P(all girls or all boys)47. P(at least 3 girls) 48. P(at least 4 girls or at least 4 boys)49. P(at least 2 boys)

For questions 50-16, Then find the probability.50. There are 3 literature books, 4 algebra books, and 2 biology books on a shelf. If a book is randomly selected, what is the probability of selecting a literature books or an algebra book?

51. A die is rolled. What is the probability of rolling a 5 or a number greater than 3?

52. In the Math Club, 7 of the 20 girls are seniors, and 4 of the 14 boys are seniors. What is the probability of randomly selecting a boy or a senior to represent the Math Club at a statewide math contest?

53. A card is drawn from a standard deck of cards. What is the probability of drawing an ace or a face card? (Hint: A face cards is a jack, queen, or king).

54. One tile with each letter of the alphabet is placed in a bag, and one is drawn at random. What is the probability of selecting a vowel or a letter from the word equation?

55. Each of the numbers from 1 to 30 is written on a card and placed in a bag. If one card is drawn at random, what is the probability that the number is a multiple of 2 or a multiple of 3?

In 56-58, Find S3.

56. 1 + 2 + 3 + ….. + n =

57. 1 + 22 + 32 + …. + n2 =


58. 5 + 25 + 125 +…..+ 5n =

In 59-61, Use math induction to prove the give statements.

59. 1 + 2 + 3 + ….. + n =

60. 1 + 22 + 32 + …. + n2 =

61. 5 + 25 + 125 +…..+ 5n =


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Chapter 10 Notes: Sequences, Induction, and Probabilityteachers.sduhsd.net/chayden/documents/Chapter10Note... · Web viewA finite sequence has a limited number of terms. An infinite