A geometric series is the sum of terms of a geometric sequence.

## Example: Summing a geometric series

A geometric sequence begins:

The sum of the first seven terms is: .

Again, because there are only seven terms, it isn’t unreasonable to go ahead and add them together. But let’s do it in a way that can be generalised into a formula:

First, multiply both sides by the common ratio which is .

Then we subtract from :

Notice that on the left hand side, we just have , and on the right hand side all but two of the terms cancel out.

Simplifying to:

This is a tidy calculation to preform: we find that .

## Generalise: the formula to calculate the sum of a geometric series

To generalize, let’s write the first seven terms of a geometric sequence:

Notice that the first term is not multiplied by and that the last term is the term.

Now, if we multiply both sides by the common ratio we have:

Subtracting like we did in the example above we have

Again, all but two terms on the right hand side cancel out.

Let’s divide through by :

Now we take out the common factor on the numerator:

Finally, we are using 7 terms but we could be adding any number of terms, say terms. Let’s replace the 7 with :

And there is the formula.

Notice that to use this formula, it is not necessary to write out the terms of the sequence. The information required is , the number of terms; the first term and the common ratio.

## Practice

Find eleven questions to practice this math at the end of this mathisfun page.