Question

Let $S$ be the sum, $P$ be the product and $R$ be the sum of the reciprocals of $3$ terms of a $G.P$. Then $P^2{R}^3:S^3$ is equal to

• A. $1:1$
• B. (common ratio)${}^n:1$
• C. (first term)${}^2:$ (common ratio)${}^2$
• D. None of these

Answer 1

Answer: (A)

Let the three terms of $G.P$. be $\frac{a}{r},a,ar$.
Then $S=\frac{a}{r}+a+ar$
$=\frac{a\left(r^2+r+1\right)}{r}$
$P=a^3,R$
$=\frac{r}{a}+\frac{1}{a}+\frac{1}{ar}$
$=\frac{1}{a}\left(\frac{r^2+r+1}{r}\right)$
Now, $\frac{P^2{R}^2}{S^3}$
$=\frac{a^6\cdot \frac{1}{a^3}{\left(\frac{r^2+r+1}{r}\right)}^3}{a^3{\left(\frac{r^2+r+1}{r}\right)}^3}=1$
Therefore, the ratio is $1:1$.