If $n$ geometric means between $a$ and $b$ be ${G_1},\;{G_2},\;.....$${G_n}$ and a geometric mean be $G$, then the true relation is

If sum of infinite terms of a $G.P.$ is $3$ and sum of squares of its terms is $3$, then its first term and common ratio are

Insert three numbers between $1$ and $256$ so that the resulting sequence is a $G.P.$

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

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

The $6^{th}$ term of a $G.P.$ is $32$ and its $8^{th}$ term is $128$, then the common ratio of the $G.P.$ is