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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

in detail.

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.

 


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