For what values of $x$, the numbers $\frac{2}{7}, x,-\frac{7}{2}$ are in $G.P.$?

Let $\left\{a_k\right\}$ and $\left\{b_k\right\}, k \in N$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $a_1=b_1=4$ and $r_1 < r_2$. Let $c_k=a_k+k, \in N$. If $c_2=5$ and $c_3=13 / 4$ then $\sum \limits_{k=1}^{\infty} c_k - \left(12 a _6+8 b _4\right)$ is equal to

If $1 + \cos \alpha + {\cos ^2}\alpha + .......\,\infty = 2 - \sqrt {2,} $ then $\alpha ,$ $(0 < \alpha < \pi )$ is

If three successive terms of a$G.P.$ with common ratio $r(r>1)$ are the lengths of the sides of a triangle and $[\mathrm{r}]$ denotes the greatest integer less than or equal to $r$, then $3[r]+[-r]$ is equal to :

If $a,\;b,\;c$ are in $A.P.$, then ${3^a},\;{3^b},\;{3^c}$ shall be in

The value of ${a^{{{\log }_b}x}}$, where $a = 0.2,\;b = \sqrt 5 ,\;x = \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + .........$to $\infty $ is