If five $G.M.s$ are inserted between $486$ and $2/3$,then the fourth $G.M.$ will be:

The sum of $3$ numbers in geometric progression is $38$ and their product is $1728$. The middle number is

How many terms of the $G.P.$ $3, \frac{3}{2}, \frac{3}{4}, \ldots$ are needed to give the sum $\frac{3069}{512} ?$

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

Find the fractional value of $0.125125125 \dots$

For the functions $f(\theta) = \alpha \tan^2 \theta + \beta \cot^2 \theta$ and $g(\theta) = \alpha \sin^2 \theta + \beta \cos^2 \theta$,where $\alpha > \beta > 0$,let $\min_{0 < \theta < \pi/2} f(\theta) = \max_{0 < \theta < \pi} g(\theta)$. If the first term of a $G$.$P$. is $(\frac{\alpha}{2\beta})$,its common ratio is $(\frac{2\beta}{\alpha})$ and the sum of its first $10$ terms is $\frac{m}{n}$,where $\gcd(m,n)=1$,then $m+n$ is equal to . . . . . . .