the sum is given by
The following applets helps to see this geometrically.
In the main panel , the rectangle OABC has height a and base
. Its area is the
sum of the geometric series .
The diagonal AC has equation
The value of a and
can be changed by moving the respective sliders . For
n = 0 , the area of a trapezium with radius is parallel to y-axis and base
a is shown for n = 1 , this is added
to the trapezium with base points
a , a ( 1 + r )
giving the total blue area.
After n steps, the area of the blue trapezium will be 
and so on
...
