The value of ${4^{1/3}}{.4^{1/9}}{.4^{1/27}}...........\infty $ is

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

Let $\alpha$ and $\beta$ be the roots of $x^{2}-3 x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^{2}-6 x+q=0 .$ If $\alpha$ $\beta, \gamma, \delta$ form a geometric progression. Then ratio $(2 q+p):(2 q-p)$ is 

Let $C_0$ be a circle of radius $I$ . For $n \geq 1$, let $C_n$ be a circle whose area equals the area of a square inscribed in $C_{n-1} .$ Then, $\sum \limits_{i=0}^{\infty}$ Area $\left(C_i\right)$ equals

In a $G.P.,$ the $3^{rd}$ term is $24$ and the $6^{\text {th }}$ term is $192 .$ Find the $10^{\text {th }}$ term.

If $x, {G_1},{G_2},\;y$ be the consecutive terms of a $G.P.$, then the value of ${G_1}\,{G_2}$ will be