Geometric Series

A sequence is a list of numbers. A series is created by adding terms in the sequence. This lesson assumes that you know about geometric sequences, how to find the common ratio and how to find an explicit formula. You may want to review the basics of geometric sequences or finding formulas.

There are two ways to indicate that you are adding terms in a sequence. One is by using summation notation and one is by using subscript notation similar to how we write explicit forms of sequences.

Summation notation is explained in conjunction with arithmetic series. Summation notation can be used with geometric sequences or any sequence that can be expressed in explicit form. However, in this lesson we will focus strictly on the summation formula for geometric series.

Summary

To find the sum of terms in a geometric sequence, use the following formula. In this formula: Sn is the sum of the first n terms in a sequence a1 is the first term in the sequence r is the common ratio in the geometric sequence n is the number of terms you are adding up

If you have been working with arithmetic series you will notice one major difference in the formulas. For geometric series you do not have to know the nth term which means that not as much work is required for finding sums of geometric series.

Another major difference can be seen in the number of terms that you add up. Generally speaking, we will be adding up fewer terms in geometric series.

1. Find S₆ for the sequence .

The formula says we need to know a₁, n, and r.

Since we are being asked to find S₆, n is 6 .

To find a₁ and r, we go back and look at the general explicit formula for a geometric sequence

By comparing that to the formula given in this problem, we find a₁= 3 and r = 4.

So now substituting these values into the equation

2. Find the sum of the first 10 terms of the sequence 0.625, 3.125, 15.625, 78.125, ...

The formula says we need to know a₁, n, and r.

Since we are being asked to find the sum of the first 10 terms, n is 10.

To find r, can divide consecutive terms and find r = 3.125/0.625 = 5.

The first term listed is a₁ which has the value 0.625.

Now substituting these values into the equation