If $1\, + \,\sin x\, + \,{\sin ^2}x\, + \,...\infty \, = \,4\, + \,2\sqrt 3 ,\,0\, < \,x\, < \,\pi $ then

The greatest integer less than or equal to the sum of first $100$ terms of the sequence $\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \ldots$ is equal to

Let $\mathrm{ABC}$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $\mathrm{ABC}$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then:

If ${\log _x}a,\;{a^{x/2}}$ and ${\log _b}x$ are in $G.P.$, then $x = $

Let $a_{n}$ be the $n^{\text {th }}$ term of a G.P. of positive terms.

If $\sum\limits_{n=1}^{100} a_{2 n+1}=200$ and $\sum\limits_{n=1}^{100} a_{2 n}=100,$ then $\sum\limits_{n=1}^{200} a_{n}$ is equal to 

Find a $G.P.$ for which sum of the first two terms is $-4$ and the fifth term is $4$ times the third term.