If the fifth term of a $G.P.$ is $2$,then the product of its first $9$ terms is:

If $1+\sin x+\sin ^{2} x+\ldots$ up to $\infty = 4+2 \sqrt{3}$,where $0 < x < \pi$ and $x \neq \frac{\pi}{2}$,then $x$ is equal to

Which term of the following sequence: $2, 2\sqrt{2}, 4, \ldots$ is $128$ (in $^{\text{th}}$)?

The coefficient of $x^{49}$ in the expansion of $(x - 1)(x - \frac{1}{2})(x - \frac{1}{2^2}) \dots (x - \frac{1}{2^{49}})$ is equal to

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

$A$ person has $2$ parents,$4$ grandparents,$8$ great-grandparents,and so on. Find the total number of his ancestors during the $10$ generations preceding his own.