Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from

Let a be the first term and r be the common ratio of the G.P.

\( s_n=\dfrac{a(1-r^n)}{1-r}\)

Since there are n terms from (n +1)^th to (2n)^th term,

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

\( s_n=\dfrac{a_{n+1}(1-r^n)}{1-r}\)

a ^n +1 = ar ^{n + 1 – 1} = arⁿ

Thus, required ratio =

\( \dfrac{a(1-r^n)}{1-r}\times \dfrac{1-r}{ar^n(1-r^n)}\)

= \( \dfrac{1}{r^n}\)

Thus, the ratio of the sum of first n terms of a G.P. to the sum of terms from 

(n + 1)^th to (2n)^th term is \( \dfrac{1}{r^n}\)