If $$S, P$$  and $$R$$ are the sum, product and sum of the reciprocals of $$n$$ terms of an increasing G.P. respectively and $${S^n} = {R^n}.{P^k},$$   then $$k$$ is equal to

Solution :
$$\eqalign{ & S = \frac{{a\left( {1 - {r^n}} \right)}}{{1 - r}},P = {a^n} \cdot {r^{\frac{{n\left( {n - 1} \right)}}{2}}} \cr & R = \frac{1}{a} + \frac{1}{{ar}} + \frac{1}{{a{r^2}}} + .....\,n{\text{ terms}} = \frac{{1 - {r^n}}}{{a\left( {1 - r} \right){r^{n - 1}}}} \cr & {S^n} = {R^n}{P^k} \cr & \Rightarrow {\left( {\frac{S}{R}} \right)^n} = {P^k} \cr & \Rightarrow {\left( {{a^2}{r^{n - 1}}} \right)^n} = {P^k} \cr & \Rightarrow {P^2} = {P^k} \cr & \Rightarrow k = 2 \cr} $$