Question

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

Answer 1

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

Sum from (n+1)th to 2nth terms is $=S_{2n}-S_n$

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

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

Ratio is given by,

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

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

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