1. Geometric Sequences
A geometric sequence (or geometric progression) is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero real number, called the common ratio. The sequence discussed in the last slide,
is an example of a geometric sequence in which the first term is 1 and the common
ratio is 2.

2. Geometric Sequences
Notice that if we divide any term after the first term by the preceding term, we obtain the common ratio r = 2.

3. Geometric Sequences
If the common ratio of a geometric sequence is r, then by the definition of a geometric sequence,
Common ratio r
for every positive integer n. Therefore, we find the common ratio by choosing
any term except the first and dividing it by the preceding term.

4. nth Term of a Geometric Sequence
In a geometric sequence with first term a1 and common ratio r, the nth term, an, is given by

5. FINDING THE nth TERM OF A GEOMETRIC SEQUENCE
Example 1
Use the formula for the nth term of a geometric sequence to answer the first question posed at the beginning of this section. How much will be earned on day 20 if daily wages follow the sequence 1, 2, 4, 8, 16,…cents?
Solution

6. FINDING TERMS OF A GEOMETRIC SEQUENCE
Example 3
Find r and a1 for the geometric sequence with third term 20 and sixth term 160.
Solution
Use the formula for the nth term of a geometric sequence.
For n = 3, a3 = a1r2 = 20.
For n = 6, a6 = a1r5 = 160.

7. FINDING TERMS OF A GEOMETRIC SEQUENCE
Example 3
Find r and a1 for the geometric sequence with third term 20 and sixth term 160.
Solution
Since
Substitute this value for a1 in the second equation.
Quotient rule for exponents
Divide by 20.
Take cube roots.
Substitute.

8. FINDING TERMS OF A GEOMETRIC SEQUENCE
Example 3
Find r and a1 for the geometric sequence with third term 20 and sixth term 160.
Solution
Since
Substitute.
Divide by 4.

9. Geometric Series
A geometric series is the sum of the terms of a geometric sequence. In applications, it may be necessary to find the sum of the terms of such a sequence. For example, a scientist might want to know the total number of insects in four generations of the population discussed in Example 4.

10. Sum of the First n Terms of a 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

11. Geometric Series
We can use a geometric series to find the total fruit fly population in Example 4 over the four-generation period. With n = 4, a1 =100, and
r = 1.5,
which agrees with our previous result.

12. Infinite Geometric Series
We extend our discussion of sums of sequences to include infinite geometric sequences such as
with first term 2 and common ratio ½. Using the formula for Sn gives the following sequence of sums.

13. Solution SUMMING THE TERMS OF AN INFINITE GEOMETRIC SERIES Example 7
Evaluate
Solution
Use the formula for the sum of the first n terms of a geometric sequence to obtain
and, in general,
Let a1 = 1, r = 1/3.

14. SUMMING THE TERMS OF AN INFINITE GEOMETRIC SERIES
Example 7
The table shows the value of (1/3)n for larger and larger values of n. n 1 10
100
200

15. SUMMING THE TERMS OF AN INFINITE GEOMETRIC SERIES
Example 7
As n gets larger and larger, gets closer and closer to 0. That is,
making it reasonable that
Hence,

16. This quotient, is called the sum of the
terms of an infinite geometric sequence.
The limit is often expressed as

17. 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
If │r│> 1, then the terms get larger and larger in absolute value, so there is no limit as n →∞ . Hence the terms of the sequence will not have a sum.