Suppose the sequence $a_1, a_2, a_3, \ldots$ is a n arithmetic progression of distinct numbers such that the sequence $a_1, a_2, a_4, a_8, \ldots$ is a geometric progression. The common ratio of the geometric progression is

Let $a, b$ and $c$ be in $G.P$ with common ratio $r,$ where $a \ne 0$ and $0\, < \,r\, \le \,\frac{1}{2}$.  If $3a, 7b$ and $15c$ are the first three terms of an $A.P.,$ then the $4^{th}$ term of this $A.P$ is

If $m$ is the $A.M$ of two distinct real numbers $ l$  and $n (l,n>1) $ and  $G_1, G_2$ and $G_3$ are three geometric means between  $l$ and $n$ then $G_1^4 + 2G_2^4 + G_3^4$ equals :

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, the new numbers are in $A.P.$ then the common ratio of the $G.P.$ is:

Let $0 < z < y < x$ be three real numbers such that $\frac{1}{ x }, \frac{1}{ y }, \frac{1}{ z }$ are in an arithmetic progression and $x$, $\sqrt{2} y, z$ are in a geometric progression. If $x y+y z$ $+z x=\frac{3}{\sqrt{2}} x y z$, then $3(x+y+z)^2$ is equal to $............$.

If $A$ is the $A.M.$ of the roots of the equation ${x^2} - 2ax + b = 0$ and $G$ is the $G.M.$ of the roots of the equation ${x^2} - 2bx + {a^2} = 0,$ then