If the third term of a $G.P.$ is $4$ then the product of its first $5$ terms is

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 in an infinite $G.P.$ first term is equal to the twice of the sum of the remaining terms, then its common ratio is

The geometric series $a + ar + ar^2 + ar^3 +..... \infty$ has sum $7$ and the terms involving odd powers of $r$ has sum $'3'$, then the value of $(a^2 -r^2)$ is -

Which term of the following sequences:

$\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots$ is $\frac{1}{19683} ?$

The sum of the first $n$ terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{{15}}{{16}} + .........$ is