* Find the common ratio for a geometric sequence. * List the terms of a geometric sequence. * Use a recursive formula for a geometric sequence. * Use an explicit formula for a geometric sequence.

1. 9.3 Geometric Sequences
Chapter 9 Sequences, Probability, and Counting
Theory

2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Find the common ratio for a geometric sequence.
⚫ List the terms of a geometric sequence.
⚫ Use a recursive formula for a geometric sequence.
⚫ Use an explicit formula for a geometric sequence.

3. Geometric Sequences
⚫ A geometric sequence is a sequence in which each term
equals a constant multiplied by the preceding term.
⚫ The constant for a geometric sequence is called the
common ratio, r, because the ratio between any two
adjacent terms equals this constant.
⚫ Like arithmetic sequences, formulas for calculating an for
geometric sequences can be found by linking the term
number to the term value.

4. Geometric Sequences (cont.)
⚫ Consider the geometric sequence:
3, 6, 12, 24, 48, …

5. Geometric Sequences (cont.)
⚫ Consider the geometric sequence:
3, 6, 12, 24, 48, …
This sequence has a1 = 3 and common ratio r = 2. Thus:
=
1 3
a
= 
2 3 2
a
=   =  2
3 3 2 2 3 2
a
=    =  3
4 3 2 2 2 3 2
a
−
=  1
3 2n
n
a

6. Geometric Sequences (cont.)
⚫ The nth term of a geometric sequence equals the first
term multiplied by (n – 1) common ratios. That is,
⚫ A geometric sequence is actually just an example of an
exponential function. The only difference is that the
domain of a geometric sequence is  rather than all real
numbers.
−
= 1
1
n
n
a a r

7. Examples
1. Calculate a100 for the geometric sequence with first
term a1 = 35 and common ratio r = 1.05.

8. Examples
1. Calculate a100 for the geometric sequence with first
term a1 = 35 and common ratio r = 1.05.
( )( )
−
= 100 1
100 35 1.05
a
( )( )
99
35 1.05 4383.375262
= =

9. Examples
2. Write an explicit formula for the nth term of the
following geometric sequence:
{2, 10, 50, 250, …}

10. Examples
2. Write an explicit formula for the nth term of the
following geometric sequence:
{2, 10, 50, 250, …}
The first term, a1, is 2. The common ratio can be found
by dividing the second term by the first term:
10
5
2
r = =

11. Examples
2. Write an explicit formula for the nth term of the
following geometric sequence:
{2, 10, 50, 250, …}
The nth term can be found by plugging these values into
the formula:
( )
1
2 5n
n
a −
=

12. Classwork
⚫ College Algebra 2e
⚫ 9.3: 6-16 (even); 9.2: 26-42 (even); 9.1: 40-50 (even)
⚫ 9.3 Classwork Check
⚫ Quiz 9.2