Find the Geometric Mean $(G.M.)$ of the sequence $1, 2, 2^2, \dots, 2^n$.

Let $S_1, S_2, \dots$ be squares such that for each $n \ge 1$,the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10 \ cm$,then for which of the following values of $n$ is the area of $S_n$ less than $1 \ cm^2$?

If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is

The number of ways of selecting $3$ numbers that are in $G.P.$ from the set $\{1, 2, 3, \ldots, 100\}$ is

The interior angles of an $n$-sided convex polygon are in $G.P.$ The smallest angle is $1^\circ$ and the common ratio is $2$. Then the number of possible values of $n$ 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 $.........$.