If $x, G_1, G_2, y$ are the consecutive terms of a $G.P.$,then the value of $G_1 G_2$ is

If $1+\sin x+\sin ^{2} x+\ldots$ up to $\infty = 4+2 \sqrt{3}$,where $0 < x < \pi$ and $x \neq \frac{\pi}{2}$,then $x$ is equal to

Let $A_{1}, A_{2}, A_{3}, \ldots$ be an increasing geometric progression of positive real numbers. If $A_{1} A_{3} A_{5} A_{7} = \frac{1}{1296}$ and $A_{2} + A_{4} = \frac{7}{36}$,then the value of $A_{6} + A_{8} + A_{10}$ is equal to

If $y = x^{1/3} \cdot x^{1/9} \cdot x^{1/27} \cdot \dots \infty$,then $y = \dots$

The sum of the first four terms of a geometric progression $(G.P.)$ is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}$. If the product of the first three terms of the $G.P.$ is $1$,and the third term is $\alpha$,then $2\alpha$ is ....... .

If $p, q, r$ are in one geometric progression and $a, b, c$ are in another geometric progression,then $cp, bq, ar$ are in