Question

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

• A. $1$
• B. $2$
• C. $3$
• D. None of these

Answer 1

Answer: (B)

$S=\frac{a\left(1-r^{n}\right)}{1-r}, P=a^{n}.r ^{\frac{n\left(n-1\right)}{2}}$
$R=\frac{1}{a}+\frac{1}{ar}+\frac{1}{ar^{2}}+.....n \,terms=\frac{1-r^{n}}{a\left(1-r\right)r^{n-1}}$
$S^{n}=R^{n}P^{k} \Rightarrow \left(\frac{S}{R}\right)^{n}=P^{k}$
$\Rightarrow \left(a^{2}\,r^{n-1}\right)^{n}=P^{k}
\Rightarrow P^{2}=P^{k} \Rightarrow k=2$