### Geometric progression

In a **geometric progression** (or geometric sequence) each successive element emerges from its predecessor by multiplication with a constant. If *a* is the first element of the series and *r* the constant, then the whole series is defined. For |*r*| < 1 the sum is

##### Explanation

Then both

as well as

We subtract the second expression from the first and find

so

And so

For |*r*| < 1 and *n*→∞ you get

##### Example 1

We investigate the infinite series

with a drawing.

1/4 1/16 1/32 1/8 1/2

The terms are

so applies

and

The sum is calculated as

and that is exactly 1.