1. 9.3 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.
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.

2. 9.3 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 since

3. 9.3 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

4. 9.3 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. 9.3 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 = Using n=4 in the formula
for an gives
or about 338 insects in the fourth generation.

6. 9.3 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:

7. 9.3 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
where

8. 9.3 Finding the Sum of the First n Terms
Example Find
Solution This is the sum of the first six terms of a
geometric series with and r = 3.
From the formula for Sn , .

9. 9.3 Infinite Geometric Series
If a1, a2, a3, … is a geometric sequence and the
sequence of sums S1, S2, S3, …is a convergent
sequence, converging to a number S. Then S is
said to be the sum of the infinite geometric series

10. 9.3 An Infinite Geometric Series
Given the infinite geometric sequence
the sequence of sums is S1 = 2, S2 = 3, S3 = 3.5, …
The calculator screen shows
more sums, approaching a value
of 4. So

11. 9.3 Infinite Geometric Series
Sum of the Terms of an Infinite Geometric Sequence
The sum of the terms of an infinite geometric sequence with first term a1 and common ratio r, where –1 < r < 1 is given by .

12. 9.3 Finding Sums of the Terms of Infinite Geometric Sequences
Example Find
Solution Here and so .