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Q.
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
Sequences and Series
Solution:
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 $1 : 1$.