We know that the terms in G.P. are:
a, ar, ar^{2}, ar^{3}, ar^{4}, ................, ar^{n1}
Let s be the sum of these terms, then
s = a + ar + ar^{2} + ar^{3} + ar^{4} + ................+ ar^{n1}
or
s

= 

This is obtained as follows:
We know that
s = a + ar + ar^{2} + ar^{3} + ar^{4} +.............+ ar^{n  1} ......(1)
Multiplying this with "r" throughout, we have
r.s = r.a + r.ar + r.ar^{2} + r.ar^{3} + r.ar^{4} +........+ r.ar^{n1}
= ar + ar^{2} + ar^{3} + ar^{4} + ar^{5} +.......+ ar^{n} ......(2)
Subtracting (1) from (2), we have
r.s  s = (ar  a) + (ar^{2}  ar) + (ar^{3}  ar^{2}) +.......+ (ar^{n}  ar^{n1})
After canceling the terms equal in magnitude but opposite in sign, we are left with
s(r  1) 
= 
ar^{n}  a 
s(r  1) 
= 
a(r^{n}  1) 
or s 
= 

By changing the signs in the numerator and the denominator we can also write the above equation as
s 
= 

What happens to the above formula if the value of n is very large. The above formula can be written as
s

= 



As the value of n approaches infinity (very large) the expression becomes smaller to that extent where we ignore it. In this case the nth term is given as
T_{n}

= 

Now we look at a couple of examples.
Example
Find the sum of the series which is given below to 13 terms.
81, 54, 36, .............
The first term 'a' = 81 and the common ratio is obtained from the ratio of 54 and 81 or 36 and 54. It is 54/81 = 2/3. Now we employ the formula given above to calculate the sum of series to 13 terms.
s 
= 





= 241.78
The same series if considered as an infinite series, the sum of n terms would be
T 
= 



= 
243 