Warm up 1. Determine if the sequence is

arithmetic. If it is, find the common difference.• 35, 32, 29, 26, ...

2. Given the first term and the common difference of an arithmetic sequence find the first five terms and the explicit formula.• a1 = 28, d = 10

Lesson 12-2 Geometric Sequences & Series

Objective: To find the nth term and the geometric means of a

geometric sequenceTo find the sum of n terms of a

geometric series

Geometric sequences increase by a constant factor called the common ratio (r)• an = ran-1

Geometric Sequences are also called geometric progressions.

Geometric Sequence

Common ratio

1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. The common ratio (r), is 2

Geometric Sequences and Series

A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.

Finding the Common Ratio

In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it.

The geometric sequence 2, 8, 32, 128, …has common ratio r = 4 since8 32 128

... 42 8 32

Geometric Sequences and Series

nth Term of a Geometric Sequence

In the geometric sequence with first term a1 and common ratio r, the nth term an, is

11n

na a r

Using the Formula for the nth Term

Example Find a5 and an for the geometric

sequence 4, –12, 36, –108 , …

Solution Here a1= 4 and r = 36/ –12 = – 3. Using

n=5 in the formula

In general

5 1 45 4 ( 3) 4 ( 3) 324a

1 11 4 ( 3)n n

na a r

11n

na a r

Modeling a Population of Fruit Flies

Example A population of fruit flies grows in such a

way that each generation is 1.5 times the previous

generation. There were 100 insects in the first generation. How many are in the fourth

generation.

Solution The populations form a geometric sequence

with a1= 100 and r = 1.5 . Using n=4 in the formula

for an gives

or about 338 insects in the fourth generation.

3 34 1 100(1.5) 337.5a a r

Geometric Means

In a geometric sequence the terms between two nonconsecutive terms are called geometric means.

Practice Write a sequence that has two

geometric means between 128 and 54.

Find the common ratio.

Use r to find the other terms. 128, 96,72,54

312854 r3

128

54r

3

64

27r

3 33

64

27r

r4

3

11n

na a r

Geometric Series

A geometric series is the sum of the terms of a geometric sequence .

In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series:

1 2 3 4

2 3100 100(1.5) 100(1.5) 100(1.5)

813

a a a a

Geometric Sequences and Series

Sum of the First n Terms of an Geometric Sequence

If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by

or

where .

1(1 )

1

n

n

a rS

r

1r

r

raaS

n

n

1

11

Finding the Sum of the First n Terms

Example Find the sum of the first six terms of a geometric series if a1 = 6 and r=3.

Solution 6

6

6(1 3 ) 6(1 729) 6( 728)2184

1 3 2 2S

Geometric vs. arithmetic sequences

The difference is in how they grow

Arithmetic sequences increase by a constant amount• an = 3n• The sequence {an} is { 3, 6, 9, 12, … }• Each number is 3 more than the last• Of the form: f(x) = dx + a

Geometric sequences increase by a constant factor• bn = 2n

• The sequence {bn} is { 2, 4, 8, 16, 32, … }• Each number is twice the previous• Of the form: f(x) = arx

Practice Determine if the sequence is

geometric. If it is, find the common ratio.

1) −1, 6, −36, 216, ... 2) −1, 1, 4, 8, ...

Given the first term and the common ratio of a geometric sequence find the first five terms and the explicit formula.

3)a1= 0.8, r = −5 4) a1= 1, r = 2