If $\alpha, \beta, \gamma$ are the geometric means between $ca, ab$; $ab, bc$; and $bc, ca$ respectively,where $a, b, c$ are in $A.P.$,then $\alpha^2, \beta^2, \gamma^2$ are in

The sum of $n$ terms of the series $\frac{1}{1 + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{7}} + \dots$ is

If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that the $4^{\text{th}}$ $A.M.$ is equal to the $2^{\text{nd}}$ $G.M.$,then $m$ is equal to:

Three numbers form a $G.P.$ If the $3^{rd}$ term is decreased by $64$,the three numbers thus obtained will constitute an $A.P.$ If the second term of this $A.P.$ is decreased by $8$,a $G.P.$ will be formed again. Find the numbers.

If the angles of a quadrilateral are in $A.P.$ whose common difference is $10^o$,then the angles of the quadrilateral are

If $S$ is the sum to infinity of a $G.P.$,whose first term is $a$,then the sum of the first $n$ terms is