1. Geometric Sequences & Series
Section 8.3
Geometric
Sequences
&
Series

2. Geometric Sequences & Series
A sequence in which a set number is multiplied to each
previous term is called a geometric sequence.
Definition of Geometric Sequence
A sequence is geometric if the ratios of consecutive
terms are the same. So, the sequence
a1, a2, a3, a4, …, an, … is geometric if there is a number
r such that
The number r is the common ratio of the sequence.

3. Geometric Sequences & Series
Examples:
Determine if each of the following is a geometric sequence.
If so, determine the common ratio.

4. Geometric Sequences & Series
How does one find a certain term of a geometric sequence?
The nth term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1rn–1
where r is the common ratio of consecutive terms of the
sequence.

5. Geometric Sequences & Series
Examples:
1. What are the first five
terms of the geometric
sequence whose first term
is a1 = 2 and whose common
ratio is r = 3.
2. Find the 10th term of the
geometric sequence whose
first term is 8 and whose
common ratio is ¼ .

6. Geometric Sequences & Series
Examples:
4. Find the 8th term of the
geometric sequence whose
third term is 1/25 and whose
sixth term is 1/3125.
Find a formula for the
nth term of the following
sequence:
4, 12, 36, 108, 324,…

7. Geometric Sequences & Series
The Sum of a Finite Geometric Sequence
The Sum of a Finite Geometric Sequence
The sum of the finite geometric sequence
a1, a1r, a1r2, a1r3, a1r4, …, a1rn–1
with common ratio r  1 is given by

8. Geometric Sequences & Series
Examples:
Find the sum

9. Geometric Sequences & Series
How does one find the sum of an infinite geometric series?
The Sum of an Infinite Geometric Series
If |r| < 1, then the infinite geometric series
a1 + a1r + a1r2 + a1r3 +… + a1rn–1 + …
has the sum

10. Geometric Sequences & Series
Example:
Find the sum of the infinite geometric series defined by
Remember to use the
formula for an infinite
geometric series. Why?

11. Geometric Sequences & Series
Example:
Brandon deposits $100 at the beginning of each month into an
account that pays 6% interest compounded monthly. What is
the balance at the end of 3 years?
Remember the first
deposit will earn
interest for 36 months,
but the last deposit only
earns interest for one
month which is a1.

12. Geometric Sequences & Series
What you should know:
1. How to find the nth term of a geometric sequence.
2. How to find the partial sum of a geometric sequence
(applying the sum formula for a finite geometric series).
3. How to find the sum of an infinite geometric series.