If $4^{th}$ and $8^{th}$ terms of a $G.P.$ are $24$ and $384$ respectively,then find out the first term and common ratio.

The first two terms of a geometric progression add up to $12.$ The sum of the third and the fourth terms is $48.$ If the terms of the geometric progression are alternately positive and negative,then the first term is

The sum of the series $\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \dots$ up to infinity is:

The sum to infinity of a geometric progression is $4/3$ and the first term is $3/4$. The common ratio is

$A$ person is to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \ldots = a_{10} = 150$ and $a_{10}, a_{11}, \ldots$ are in an $A.P.$ with common difference $-2$,then the time taken by him to count all notes is ............... minutes.

If ${A_1}, {A_2}$; ${G_1}, {G_2}$ and ${H_1}, {H_2}$ are $AMs$,$GMs$,and $HMs$ between two quantities,then the value of $\frac{{G_1 G_2}}{{H_1 H_2}}$ is