Copyright © 2007 Pearson Education, Inc. Slide 8-1

Geometric Sequences

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 multiplier from each term to the next is called the common ratio and is usually denoted by r.

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.

Copyright © 2007 Pearson Education, Inc. Slide 8-2

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 1, 2, 4, 8, 16 , …

has common ratio r = 2 since:

Copyright © 2007 Pearson Education, Inc. Slide 8-3

Writing a Recursive Formula for a geometric sequence The recursive formula is used to

find the next term in the sequence

The recursive formula would just be as follows: the next term in the sequence (an) is equal to the previous term (an-1) times the common ratio (r)

What would the recursive formulas for the following sequences be?(fill in the first term and the common ratio into the recursive formula format)

a. 9, 81, 729b. 81, 27,9 (hint: the common ratio will be a fraction)

a1 = first

term

an= an-1 *r

Copyright © 2007 Pearson Education, Inc. Slide 8-4

Writing an Explicit Formula for a geometric sequence

The explicit formula is used to find any term in the sequence

The explicit formula would just be as follows:

the term that you are solving for (an) is equal to the first term(a1) times the common ratio (r)

to the power of the previous term number

an= a1r(n-1)

What would the explicit formulas for the following sequences be?(fill in the first term (a1) and the common ratio (r) into the recursive formula format)

a. 4, -8, 16… (r is negative) b. 125, 25, 5… (r is a fraction)

Copyright © 2007 Pearson Education, Inc. Slide 8-5

Using an Explicit Formula for Finding the nth Term

Example Find a5 for the geometric

sequence 4, –12, 36, –108 , …

first term = 4

Common ratio = -3

Finding the Explicit Formula:

an= a1r(n-1)

an= (4)(-3)(n-1)

Plugging in to find a5:

a5= (4)(-3)(5-1)

a5= 4 x (-3)4

a5 = 4 x 81

a5 = 324

You do: Find a6 for the

following geometric sequence:

4, -8, 16…

Copyright © 2007 Pearson Education, Inc. Slide 8-6

Geometric Sequence Word ProblemExample 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.

common ratio = 1.5

first term = 100

nth term = 4

Solution

Finding the Explicit Formula: an= a1r(n-1)

an= (100)(1.5)(n-1)

Plugging in to find a4:a4= (100)(1.5)(4-1)

a4= 100 x (1.5)3

a4 = 100 x 3.375a4 = approx. 324 flies