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

- [JEE MAIN 2020]

- A
$2^{\frac{1}{2}}$

- B
$2^{\frac{1}{4}}$

- C
$2$

- D
$1$

The sum of some terms of $G.P.$ is $315$ whose first term and the common ratio are $5$ and $2,$ respectively. Find the last term and the number of terms.

Find the $20^{\text {th }}$ and $n^{\text {th }}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$

Given a $G.P.$ with $a=729$ and $7^{\text {th }}$ term $64,$ determine $S_{7}$

The sum of $100$ terms of the series $.9 + .09 + .009.........$ will be

Let ${a_n}$ be the ${n^{th}}$ term of the G.P. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta $, such that $\alpha \ne \beta $,then the common ratio is

- [IIT 1992]