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

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

if $x = \,\frac{4}{3}\, - \,\frac{{4x}}{9}\, + \,\,\frac{{4{x^2}}}{{27}}\, - \,\,.....\,\infty $ , then $x$ is equal to

Let $M=2^{30}-2^{15}+1$, and $M^2$ be expressed in base $2$.The number of $1$'s in this base $2$ representation of $M^2$ is

If the sum of first 6 term is $9$ times to the sum of first $3$ terms of the same $G.P.$, then the common ratio of the series will be

Let $n \geq 3$ and let $C_1, C_2, \ldots, C_n$, be circles with radii $r_1, r_2, \ldots, r_n$, respectively. Assume that $C_i$ and $C_{i+1}$ touch externally for $1 \leq i \leq n-1$. It is also given that the $X$-axis and the line $y=2 \sqrt{2} x+10$ are tangential to each of the circles. Then, $r_1, r_2, \ldots, r_n$ are in