May 1, 2012 Arithmetic and Geometric Sequences Warm-up: What is the difference between an arithmetic and geometric sequence? Write an example for each.

Check HW 9.1 with your group

Lesson 9.2 Arithmetic Sequences An arithmetic sequence is a list of numbers such that the numbers go from one term to the next by adding the same value, called the common difference, d. What is the common difference (d) for the sequence 2, -3, -8, -13,… ?

Two formulas to find the nth term for an arithmetic sequence. Linear form: a n = dn + c, where c = a 1 – d Alternative form: a n = a 1 + (n – 1)d Example 1: Find a formula for a n, for the arithmetic sequence. Given a 1 = 15 and d = 4 1. Find c, using c = a 1 – d c = 15 – 4 c = Plug c and d into the linear form. a n = 4n + 11 Now find the first five terms for a n. What is another way to find the five terms?

Practice: Arithmetic Sequence a n = dn + c 1.Find the formula for 10, 5, 0, -5, -10… 2. The 4 th term of an arithmetic sequence is 20, and the 13 th term is 65.. Write the first 11 terms of this sequence. 1. Find c, using c = a 1 – d 2. Plug c and d into the linear form. *Hint: find d a n = -5n + 15 d = 5 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55,..,

Lesson 9.3 Geometric Sequences A geometric sequence is a list of numbers such that the numbers go from one term to the next by multiplying the same value, called the common ratio, r. What is the common ratio (r) for the sequence 9, -6, 4, -8/3,… ?

Example 2 Formula for Geometric Sequences a n = a 1 r n – 1 Or to find the (n + 1)th term a n+1 = ra n a) Write the first five terms of the geometric sequence if a 1 = 6 and r = 2. 6, 12, 24, 48, 96 b) Write an expression for the nth term of the geometric sequence, and find a 20. a n = 6(2) n – 1 a 20 = 6(2) 19 a 20 = 3,145,728

Practice: Write the first five terms of the geometric sequence. Determine the common ratio and write the nth term of the sequence as a function of n if a 1 = 7, a k + 1 = 2a k a n = a 1 r n – 1 a n = 7(2) n – 1