If the fifth term of a $G.P.$ is $2$,then the product of its first $9$ terms is:

Let the positive numbers $a_1, a_2, a_3, a_4$ and $a_5$ be in a $G$.$P$. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively,where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$,then $m+n$ is equal to $.........$.

If the first and the $n^{\text{th}}$ term of a $G.P.$ are $a$ and $b$ respectively,and if $P$ is the product of $n$ terms,prove that $P^{2} = (ab)^{n}$.

The $7^{th}$ term of the sequence $\sqrt{2}, \sqrt{10}, 5\sqrt{2}, \dots$ is

Given a $G.P.$ with $a=729$ and $7^{\text{th}}$ term $64,$ determine $S_{7}$.

Evaluate $\sum\limits_{k = 1}^{11} {\left( {2 + {3^k}} \right)} $