If the $n^{th}$ term of geometric progression $5, - \frac{5}{2},\frac{5}{4}, - \frac{5}{8},...$ is $\frac{5}{{1024}}$, then the value of $n$ is

If three geometric means be inserted between $2$ and $32$, then the third geometric mean will be

The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$, will be

The sum of the series $3 + 33 + 333 + ... + n$ terms is

The $G.M.$ of roots of the equation ${x^2} - 18x + 9 = 0$ is

If the first and the $n^{\text {th }}$ term of a $G.P.$ are $a$ and $b$, respectively, and if $P$ is the product of $n$ terms, prove that $P^{2}=(a b)^{n}$