Mark the correct alternative in the following question:

Solution:

Let the three terms of the G.P. be $\frac{a}{n}, a, a r$. Then

$S=\frac{a}{r}+a+a r$

$=a\left(\frac{1}{r}+1+r\right)$

$=a\left(\frac{1+r+r^{2}}{r}\right)$

$=\frac{a\left(r^{2}+r+1\right)}{r}$

Also,

$P=\frac{a}{r} \times a \times a r=a^{3}$

And,

$R=\frac{r}{a}+\frac{1}{a}+\frac{1}{a r}$

$=\frac{1}{a}\left(r+1+\frac{1}{r}\right)$

$=\frac{1}{a}\left(\frac{r^{2}+r+1}{r}\right)$

Now,

$\frac{P^{2} R^{3}}{S^{3}}=\frac{\left(a^{3}\right)^{2} \times\left[\frac{1}{a}\left(\frac{r^{2}+r+1}{r}\right)\right]^{3}}{\left[a\left(\frac{r^{2}+r+1}{r}\right)\right]^{3}}$

$=\frac{a^{6} \times \frac{1}{a^{3}}\left(\frac{r^{2}+r+1}{r}\right)^{3}}{a^{3}\left(\frac{r^{2}+r+1}{r}\right)^{3}}$

$=\frac{1}{1}$

So, the ratio is $1: 1$.

Hence, the correct alternative is option (a).