In a geometric progression,the first term is $1$. If $4T_2 + 5T_3$ is minimum,then what is the common ratio?

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 $a = \dots$

The sum of three numbers in a Geometric Progression $(GP)$ is $38$ and their product is $1728$. The largest of these numbers is.......

The sum of a few terms of a geometric series is $728$. If the common ratio is $3$ and the last term is $486$,then the first term of the series will be:

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 roots of the equation $x^5-40x^4-Px^3-Rx-S=0$ are in geometric progression and the sum of the reciprocals of the roots is $10$,then $|S|=$