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

For $0<\mathrm{c}<\mathrm{b}<\mathrm{a}$, let $(\mathrm{a}+\mathrm{b}-2 \mathrm{c}) \mathrm{x}^2+(\mathrm{b}+\mathrm{c}-2 \mathrm{a}) \mathrm{x}$ $+(c+a-2 b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements

$(I)$ If $\alpha \in(-1,0)$, then $\mathrm{b}$ cannot be the geometric mean of $\mathrm{a}$ and $\mathrm{c}$

$(II)$ If $\alpha \in(0,1)$, then $\mathrm{b}$ may be the geometric mean of $a$ and $c$

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

The sum of infinite terms of the geometric progression $\frac{{\sqrt 2 + 1}}{{\sqrt 2 - 1}},\frac{1}{{2 - \sqrt 2 }},\frac{1}{2}.....$ is

Which term of the following sequences:

$\sqrt{3}, 3,3 \sqrt{3}, \ldots$ is $729 ?$

Suppose the sides of a triangle form a geometric progression with common ratio $r$. Then, $r$ lies in the interval