Question

Let S be the sum, P be the product and R be the sum of the reciprocals of n terms of a GP. Then $P^2{R}^n:S^n$ is equal to

• A. $1:1$
• B. ${\left(commonratio\right)}^n:1$
• C. ${\left(firstterm\right)}^2:{\left(commonratio\right)}^n$
• D. none of these

Answer 1

The correct option is A $1:1$

Let the first term and common ratio of a G.P be A and R respectively.
so
$S=A\frac{R^n-1}{R-1}P=A\left(AR\right)\left(A{R}^2\right)\left(A{R}^3\right)........\left(A{R}^{n-1}\right)\Rightarrow P=A^n{R}^{(1+2+3+........+n-1)}=A^n{R}^{\left(\frac{n(n-1)}{2}\right)}andR=A^{-1}+{\left(AR\right)}^{-1}+{\left(AR\right)}^{-2}+.......+{\left(AR\right)}^{-(n-1)}\Rightarrow R=A^{-1}\frac{R^{-n}-1}{R^{-1}-1}\Rightarrow R=\frac{1-R^n}{A(1-R)R^{n-1}}$
$\frac{P^2{R}^n}{S^n}=A^{2n}{R}^{\left(n(n-1)\right)}\times \left\{\frac{{(1-R^n)}^n}{A^n{(1-R)}^n{R}^{n(n-1)}}\right\}\left\{\frac{{(R^n-1)}^n}{A^n{(R-1)}^n}\right\}=1$