Sum of the First n Terms of a Geometric Sequence
If a sequence is geometric there are ways to find the sum of the first n terms, denoted Sn, without actually adding all of the terms.
To find the sum of the first n terms of a geometric sequence use the formula
,
where n is the number of terms, a1 is the first term and r is the common ratio.
Example 1:
Find the sum of the first 8 terms of the geometric series if a1 = 1 and r = 2.
Example 2:
Find S10 of the geometric sequence 24, 12, 6,···.
First, find r.
Now, find the sum:
Example 3:
Evaluate.
(You are finding S10 for the series 3 – 6 + 12 – 24 + ···, whose common ratio is –2.)
In order for an infinite geometric series to have a sum, the common ratio r must be between –1 and 1. Then as n increases, rn gets closer and closer to 0. To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, 
, where a1 is the first term and r is the common ratio.
Example 1:
Find the sum of the infinite geometric sequence
27, 18, 12, 8,···.
First find r:
Then find the sum:
Example 2:
Find the sum of the infinite geometric sequence
8, 12, 18, 27,··· if it exists.
First find r:
Since r = 3/2 is not less than one the series has no sum.
