Sum of infinite number of terms in $G.P.$ is $20$ and sum of their square is $100$. The common ratio of $G.P.$ is

The third term of a $G.P.$ is the square of first term. If the second term is $8$, then the ${6^{th}}$ term is

If the ${4^{th}},\;{7^{th}}$ and ${10^{th}}$ terms of a $G.P.$ be $a,\;b,\;c$ respectively, then the relation between $a,\;b,\;c$ is

The $4^{\text {tht }}$ term of $GP$ is $500$ and its common ratio is $\frac{1}{m}, m \in N$. Let $S_n$ denote the sum of the first $n$ terms of this GP. If $S_6 > S_5+1$ and $S_7 < S_6+\frac{1}{2}$, then the number of possible values of $m$ is $..........$

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

If $3 + 3\alpha + 3{\alpha ^2} + .........\infty = \frac{{45}}{8}$, then the value of $\alpha $ will be