${7^{th}}$ term of the sequence $\sqrt 2 ,\;\sqrt {10} ,\;5\sqrt 2 ,\;.......$ is

Suppose four distinct positive numbers $a_1, a_2, a_3, a_4$ are in $G.P.$ Let $b_1=a_1, b_2=b_1+a_2, b_3=b_2+a_3$ and $b_4=b_3+a_4$.

$STATEMENT-1$ : The numbers $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \mathrm{~b}_4$ are neither in $A.P$. nor in $G.P.$ and 

$STATEMENT-2$ : The numbers $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \mathrm{~b}_4$ are in $H.P.$

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

Insert two numbers between $3$ and $81$ so that the resulting sequence is $G.P.$

If $G$ be the geometric mean of $x$ and $y$, then $\frac{1}{{{G^2} - {x^2}}} + \frac{1}{{{G^2} - {y^2}}} = $

The value of ${4^{1/3}}{.4^{1/9}}{.4^{1/27}}...........\infty $ is