The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$,will be

If $1 + \sin x + \sin^2 x + \dots \text{ to } \infty = 4 + 2\sqrt{3}$,where $0 < x < \pi$,then:

If $x = \sum\limits_{n = 0}^\infty {{a^n}} ,\;y = \sum\limits_{n = 0}^\infty {{b^n},\;z = \sum\limits_{n = 0}^\infty {{{(ab)}^n}} } $,where $a, b < 1$,then

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a $G$.$P$. such that $a_{1} < 0$; $a_{1} + a_{2} = 4$ and $a_{3} + a_{4} = 16$. If $\sum_{i=1}^{9} a_{i} = 4 \lambda$,then $\lambda$ is equal to:

If $x, y, z$ are in $G.P.$ and $a^x = b^y = c^z$,then

If the sum of the second,third and fourth terms of a positive term $G.P.$ is $3$ and the sum of its sixth,seventh and eighth terms is $243,$ then the sum of the first $50$ terms of this $G.P.$ is