Question

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms form $(n+1{)}^{th}$ to $(2n{)}^{th}$ term is $\frac{1}{r^n}$.

Answer 1

Sum of first n terms,

$S_n=\frac{a(1-r^n)}{1-r}$

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

$S_{2n}-S_n$

$=\frac{a(1-r^{2n})}{1-r}-\frac{a(1-r^n)}{1-r}$

$=\frac{a(r^n-r^{2n})}{1-r}$

Ratio,

$=\frac{\frac{a(1-r^n)}{1-r}}{\frac{a(r^n-r^{2n})}{1-r}}$

$=\frac{1-r^n}{r^n-r^{2n}}$

$=\frac{1-r^n}{r^n(1-r^n)}$

$=\frac{1}{r^n}$

Hence proved.