The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$,will be

The first three terms of a geometric progression are $a, b, c$. If the harmonic mean of $a$ and $b$ is $12$ and the harmonic mean of $b$ and $c$ is $36$,then $a = \dots$

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 $.........$.

The geometric mean of $n$ positive terms $x_1, x_2, \dots, x_n$ is equal to:

If $x > 1, y > 1, z > 1$ are in geometric progression,then in which progression are $\frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z}$?

The sum of the first ten terms of a geometric progression is $S_1$ and the sum of the next ten terms ($11^{th}$ to $20^{th}$) is $S_2$. What is the common ratio?