What is a geometric series?

A geometric series is a series of a geometric sequence. Given a geometric sequence equation whose equationth term is equation, we can define its geometric series equation (which is the sum of the first equation terms of the sequence) as:

    equation

According to the above formula, we need to know the first term equation, the last term equation as well as the ratio equation. We can see that this is not defined if the ratio equation is 1.

Another equation for the series is:

    equation

In this case we need to know the first term equation, the number of terms equation the ratio equation.

Example of a geometric series

Consider the following sequence:

    equation

Let us find the sum of the first 15 terms. Of course we could determine all 15 terms and then sum them up. Or we can use the above formula. We already know the first term equation and we only need to find the ratio equation to be able to use the formula. We find equation as the ratio of any two successive terms: equation. We obtain the series equation as:

    equation

Now we can easily find the sum of the first 15 terms as equation.