# 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.

The sum of the first n terms of a geometric sequence is called geometric series.

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.

There is a formula to calculate the nth term of an geometric series, that is, the sum of the first n terms of an geometric sequence.