Sum of a G.P. series is represented by the formula, , when r≠1. ‘S_n’ represents the sum of the G.P. series upto n^th terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Thus, the sum of the first n terms of the G.P. series is,

Sum of (n+1)^th term to 2n^th term

= Sum of the first 2n^th term – the sum of 1^st term to n^th term

= -

=

=

=

The ratio of the sum of first n terms of the G.P. to the sum of the terms from (n + 1)^th to (2n)^th term

=

[Cancelling out the common factors from the numerator and denominator ⇒ a, (r-1), (rⁿ – 1) ]

=

Hence Proved.