The product $2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots .$ to $\infty$ is equal to
$2^{\frac{1}{2}}$
$2^{\frac{1}{4}}$
$2$
$1$
If $1 + \cos \alpha + {\cos ^2}\alpha + .......\,\infty = 2 - \sqrt {2,} $ then $\alpha ,$ $(0 < \alpha < \pi )$ is
If $n$ geometric means between $a$ and $b$ be ${G_1},\;{G_2},\;.....$${G_n}$ and a geometric mean be $G$, then the true relation is
Find the $10^{\text {th }}$ and $n^{\text {th }}$ terms of the $G.P.$ $5,25,125, \ldots$
The sum of first three terms of a $G.P.$ is $\frac{39}{10}$ and their product is $1 .$ Find the common ratio and the terms.
If in an infinite $G.P.$ first term is equal to the twice of the sum of the remaining terms, then its common ratio is