1. 9.1-9.39.1-9.3 Sequences and Series

2. Quick Review

3. Quick Review Solutions

4. What you’ll learn about Infinite Sequences Limits of Infinite Sequences Arithmetic and Geometric Sequences Sequences and Graphing Calculators … and why Infinite sequences, especially those with finite limits, are involved in some key concepts of calculus.

5. Limit of a Sequence

6. Example Finding Limits of Sequences

8. Arithmetic Sequence

9. Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. -2, 1, 4, 7, …

10. Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. -2, 1, 4, 7, …

11. Geometric Sequence

12. Example Defining Geometric Sequences Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. 2, 6, 18,…

13. Example Defining Geometric Sequences Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. 2, 6, 18,…

14. The Fibonacci Sequence

15. 9.1-9.39.1-9.3 Sequences and Series (cont.)

16. Quick Review

17. Quick Review Solutions

18. What you’ll learn about Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergences of Geometric Series … and why Infinite series are at the heart of integral calculus.

19. Summation Notation

20. Sum of a Finite Arithmetic Sequence

21. Example Summing the Terms of an Arithmetic Sequence

23. Sum of a Finite Geometric Sequence

24. Infinite Series

25. Sum of an Infinite Geometric Series

26. Example Summing Infinite Geometric Series

28. 9.49.4 Mathematical Induction

29. Quick Review

30. Quick Review Solutions

31. What you’ll learn about The Tower of Hanoi Problem Principle of Mathematical Induction Induction and Deduction … and why The principle of mathematical induction is a valuable technique for proving combinatorial formulas.

32. The Tower of Hanoi Solution The minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2 n – 1.

33. Principle of Mathematical Induction Let P n be a statement about the integer n. Then P n is true for all positive integers n provided the following conditions are satisfied: 1. (the anchor) P 1 is true; 2. (inductive step) if P k is true, then P k+1 is true.

34. 9.59.5 The Binomial Theorem

35. Quick Review

36. Quick Review Solutions

37. What you’ll learn about Powers of Binomials Pascal’s Triangle The Binomial Theorem Factorial Identities … and why The Binomial Theorem is a marvelous study in combinatorial patterns.

38. Binomial Coefficient

39. Example Using n C r to Expand a Binomial

41. The Binomial Theorem

42. Basic Factorial Identities

43. 9.69.6 Counting Principles

44. Quick Review

45. Quick Review Solutions

46. What you’ll learn about Discrete Versus Continuous The Importance of Counting The Multiplication Principle of Counting Permutations Combinations Subsets of an n-Set … and why Counting large sets is easy if you know the correct formula.

47. Multiplication Principle of Counting

48. Example Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.

49. Example Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. You can fill in the first blank 26 ways, the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways, the sixth blank 10 ways, and the seventh blank 10 ways. By the Multiplication Principle, there are 26×26×26×26×10×10×10 = 456,976,000 possible license plates.

50. Permutations of an n-Set There are n! permutations of an n-set.

51. Example Distinguishable Permutations Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.

52. Example Distinguishable Permutations Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER. Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.

53. Distinguishable Permutations

54. Permutations Counting Formula

55. Combination Counting Formula

56. Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?

57. Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?

58. Formula for Counting Subsets of an n-Set

59. 9.79.7 Probability

60. Quick Review

61. Quick Review Solutions

62. What you’ll learn about Sample Spaces and Probability Functions Determining Probabilities Venn Diagrams and Tree Diagrams Conditional Probability Binomial Distributions … and why Everyone should know how mathematical the “laws of chance” really are.

63. Probability of an Event (Equally Likely Outcomes)

64. Probability Distribution for the Sum of Two Fair Dice OutcomeProbability 21/36 32/36 43/36 54/36 65/36 76/36 85/36 94/36 103/36 112/36 121/36

65. Example Rolling the Dice Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.

66. Example Rolling the Dice Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.

67. Probability Function

68. Probability of an Event (Outcomes not Equally Likely)

69. Strategy for Determining Probabilities

70. Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

71. Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

72. Multiplication Principle of Probability Suppose an event A has probability p 1 and an event B has probability p 2 under the assumption that A occurs. Then the probability that both A and B occur is p 1 p 2.

73. Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

74. Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

75. Conditional Probability Formula

76. Binomial Distribution

77. Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

78. Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

79. Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

80. Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?