Chapter 8 Sequences, Series & Combinations

1. Chapter 8 Sequences, Series & Combinations

2. An arithmetic sequence is one in which a number is constantly added to get the next term (they consider subtraction, adding a negative) • A geometric sequence is one in which a number is constantly multiplied to get the next term (they consider division, multiplying by the reciprocal) Section 8-1 Formulas for sequences

3. Formulas for arithmetic sequences: • Explicit formula: • Recursive formula: • You have to include the “for all…” with all recursive formulas • Formulas for geometric sequences: • Explicit formula: • Recursive formula:

4. Ex1. -3, 6, -12, 24, … • A) Write an explicit formula • B) Write a recursive formula • C) What is the 30th term? • Ex2. -6, -2, 2, 6, … • A) Write an explicit formula • B) Write a recursive formula • C) What is the 30th term?

5. Recall from chapter 6: the limiting value of the sequence is e • The way to write that is: • End behavior (in terms of sequences) describes what happens as n gets larger • Open your book to page 497 to look at the graphs at the top of the page Section 8-2 Limits of Sequences

6. The graphs are said to be “divergent” because they do not approach a limit • The second graph is said to “increase without bound” • The first graph is said to “decrease without bound” • If a sequence has a limit, it is convergent • Limit Property 1: • This will be used to find the limits of many other sequences (see graph)

7. Harmonic sequence: • Alternating harmonic seq.: • For many equations you can use algebra to rewrite them and compare them to the harmonic sequence • Ex1. • Ex2.

8. Summation notation: • This represents a finite arithmetic series (because it has an end term) • Ex1. Evaluate • Theorem: The sum of an arithmetic series with 1st term a1 and constant difference d is given by Section 8-3 Arithmetic Series

9. Use the 1st option if you know the last term, use the 2nd if you don’t • Ex2. A packer had to fill 100 boxes identically with machine tools. The shipper filled the 1st box in 13 minutes, but got faster by the same amount each time. If he filled the last box in 8 minutes, what was the total time that it took to fill the 100 boxes?

10. Ex3. A new business decides to rank its 9 employees by how well they work and pay them amounts that are in arithmetic sequence with a constant difference of $500 a year. If the total amount paid the employees is to be $250,000, what will the highest and lowest paid employees make each year?

11. Read the example of the finite geometric series for the inventor of chess on pg. 509 • The sum of the finite geometric series with first term g1 and constant ratio r ≠ 1 is given by • Ex1. Evaluate • Notice that ex. 1 is a geometric series with 1st term of 10, r = .75 and n = 6 Section 8-4 Geometric Series

12. Ex2. Suppose you have two children who marry and each of them has 2 children. Each of these offspring has 2 children and so on. If all of these progeny marry but none marry each other, and all have two children, in how many generations will you have a thousand descendants? Count your children as generation 1.

13. The sum of the infinite series is the limit of the sequence of partial sums of the series, provided the limit exists and is finite • If • If the limit exists, the series is convergent with a sum of • If the limit does not exist, the series is divergent Section 8-5 Infinite Series

14. Consider the infinite geometric series • A) If , the series converges and • B) If , the series diverges • Open your book to page 518 so we can read “An Example of an Infinite Geometric Series” • Ex1. Consider the finite geometric series below and find the first 5 partial sums

15. If a series is geometric and it’s constant is between -1 and 1, then it converges • IF a geometric series is convergent, it’s sum will be • Ex2. Find the sum of the series from Ex1.

16. Ex1. Suppose x people in a room all want to shake hands with one another (no repeats), how many handshakes will there be? • With combinations, the order does not matter • The number of combinations of n items taken r at a time can be written two ways: Section 8-6 Combinations

17. Either way, it is read “n choose r” • You can use the button on your calculator or do it with factorials • Formula: • Ex2. Twenty distinct points are chosen on a circle • A) How many segments are there with these points as endpoints? • B) How many triangles are there with these points as vertices? • C) How many quadrilaterals are there with these points as vertices?

18. Ex3. At a restaurant you can order pizza with any of 9 different toppings. How many different pizzas are possible with exactly 3 of those toppings?

19. You can see many relationships among combinations in Pascal’s triangle • Open your book to page 529 • Let n and r be nonnegative integers with r < n. The (r+1)st term in row n of Pascal’s triangle is • Ex1. Find the first 4 terms in row 8 of Pascal’s triangle • Open your book to page 530-531 and read Properties 1-5 Section 8-7 Pascal’s Triangle

20. To expand means to write a product of polynomials or a power of polynomials as a sum (if it is a binomial squared, you FOIL) • When you expand binomials raised to a power, the coefficients of the terms in the expansion rows in Pascal’s Triangle (see page 535) Section 8-8 The Binomial Theorem

21. You can expand using the binomial theorem, or you can do it the long way, but it saves time to know how to apply the binomial theorem to find the coefficient of a certain term • Ex1. Expand • Binomial Theorem: For any nonnegative integer n,

22. Ex2. Use the Binomial Theorem to expand

23. Open to page 540 (we are going to read the features of a binomial experiment) • Binomial Probability Theorem: Suppose that in a binomial experiment with n trials the probability of success is p in each trial, and the probability of failure is q, where q = 1 – p. Then P(exactly k successes) = Section 8-9 Binomial Probabilities

24. Ex1. The probability of getting a sum of 7 in a toss of two fair die is known to be • A) What is the probability of getting exactly two 7s in five tosses? • B) What is the probability of getting at least two 7s in five tosses?