We know that the terms in G.P. are:

a, ar, ar², ar³, ar⁴, ................, ar^n-1

Let s be the sum of these terms, then

s = a + ar + ar² + ar³ + ar⁴ + ................+ ar^n-1

                            or

s

=

This is obtained as follows:

We know that

s = a + ar + ar² + ar³ + ar⁴ +.............+ ar^{n - 1}                                            ......(1)

Multiplying this with "r" throughout, we have

r.s     = r.a + r.ar + r.ar² + r.ar³ + r.ar⁴ +........+ r.ar^n-1 

         = ar + ar² + ar³ + ar⁴ + ar⁵ +.......+ arⁿ                                             ......(2)

Subtracting (1) from (2), we have

r.s - s = (ar - a) + (ar² - ar) + (ar³ - ar²) +.......+ (arⁿ - ar^n-1)

After canceling the terms equal in magnitude but opposite in sign, we are left with

s(r - 1)	=	arn - a
s(r - 1)	=	a(rn - 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

Tn

=

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