1. Geometric Sequences and Series
A sequence is geometric if the ratios of consecutive terms are the same.
2, 8, 32, 128, 512, . . .
geometric sequence
The common ratio, r, is 4.
Definition of Geometric Sequence

3. 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.
a1 = 15
15, 75, 375, , . . .
a2 = 15(5)
a3 = 15(52)
a4 = 15(53)
The nth term is 15(5n-1).
The nth Term of a Geometric Sequence

5. Example 3. Find the 15th term of the geometric sequence whose first term is 20 and whose common ration is 1.05.

6. Example 4. Find a formula for the nth term of the following geometric sequence. What is the ninth term of the sequence?
5, 15, 45, …
Find the common ratio

8. upper limit of summation lower limit of summation
The sum of the first n terms of a sequence is represented by summation notation.
upper limit of summation
lower limit of summation
index of summation
Definition of Summation Notation

9. The Sum of a Finite Geometric Sequence
The sum of a finite geometric sequence is given by
= ?
n = 8
a1 = 5
The Sum of a Finite Geometric Sequence

10. Write out a few terms.
If the index began at i = 0, you would have to adjust your formula

11. Definition of Geometric Series
The sum of the terms of an infinite geometric sequence is called a geometric series.
If |r| < 1, then the infinite geometric series
a1 + a1r + a1r2 + a1r a1rn
has the sum
Definition of Geometric Series

12. Example 7. Use a graphing calculator to find the first six partial sums of the series. Then find the sum of the series.
Use the formula for the sum of an infinite series to find the sum.

13. Example 8. Find the sum of 3 + 0.3 + 0.03 + 0.003 + …,

14. Example 9. A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years?
This type of savings plan is called an increasing annuity.
The first deposit will gain interest for 24 months, and its balance will be
The second deposit will gain interest for 23 months
The last deposit will gain interest for only one month
The total balance will be the sum of the balances of the 24 deposits.