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)

$1 : 1$

Answer 2

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 $1 : 1$.