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

The $4^{\text{th}}$ term of a $GP$ is $500$ and its common ratio is $\frac{1}{m}$,where $m \in N$. 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 $..........$

If $a, b, c, d$ and $p$ are different real numbers such that $(a^{2}+b^{2}+c^{2}) p^{2}-2(ab+bc+cd) p+(b^{2}+c^{2}+d^{2}) \leq 0$,then show that $a, b, c$ and $d$ are in $G.P.$

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

The geometric mean of the sequence $1, 2, 2^2, ..., 2^n$ is:

If $\alpha, \beta$ are the roots of the equation $x^2 - 3x + a = 0$ and $\gamma, \delta$ are the roots of the equation $x^2 - 12x + b = 0$,and if $\alpha, \beta, \gamma, \delta$ form an increasing $G.P.$,then $(a, b) = $