May 1, 2012Arithmetic 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: an = dn + c, where c = a1 – d

Alternative form: an = a1 + (n – 1)d

Example 1: Find a formula for an, for the arithmetic sequence. Given a1 = 15 and d = 41. Find c, using

c = a1 – d c = 15 – 4

c = 11 2. Plug c and d into the linear form.

an = 4n + 11

Now find the first five terms for an.What is another way to find the five terms?

Practice: Arithmetic Sequence an = dn + c 1. Find the formula for 10, 5, 0, -5, -10…

2. The 4th term of an arithmetic sequence is 20, and the 13th term is 65.. Write the first 11 terms of this sequence.

1. Find c, using c = a1 – d

2. Plug c and d into the linear form.

*Hint: find d

an = -5n + 15

d = 55, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, ..,

Lesson 9.3 Geometric SequencesA 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 2Formula for Geometric Sequences

an = a1rn – 1

Or to find the (n + 1)th term an+1 = ran

a) Write the first five terms of the geometric sequence if a1 = 6 and r = 2.

6, 12, 24, 48, 96

b) Write an expression for the nth term of the geometric sequence, and find a20.

an = 6(2)n – 1 a20 = 6(2)19 a20 = 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 a1 = 7, ak + 1 = 2ak

an = a1rn – 1

an = 7(2)n – 1