Explanation

(c)
Since, $G$ is the geometric mean between $a$ and $b.$
$\therefore\ G = (ab)^{1/2}$ and $n$ geometric means $G_{1},\ G_{2},\ldots,G_{n}$ be inserted between $a$ and $b$.
$\therefore\ G_{1} = ar^{1},\ G_{2} = ar^{2},\ldots,G_{n} = ar^{n}$
Now, $G_{1} \bullet G_{2} \bullet G_{3}\ldots G_{n} = a^{n}r^{1 + 2 + ... + n}$
$= a^{n}r^{n(n + 1)/2}$
But $b = ar^{n + 1}\ \Rightarrow \ \bullet r = \left( \frac{b}{a} \right)^{\frac{1}{n + 1}}$
Therefore, the required product is
$a^{n}\left( \frac{b}{a} \right)^{\left\lbrack \frac{1}{n + 1} \right\rbrack\frac{n( + 1)}{2}} = (ab)^{\frac{n}{2}}$
$= \left\{ (ab)^{\frac{1}{2}} \right\}^{n} = G^{n}$