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.