$8^{th}$ term of the series $2\sqrt 2 + \sqrt 2 + 0 + .....$ will be

Let $S_n$ denote the sum of the first $n$ terms of an $A.P$.. If $S_4 = 16$ and $S_6 = -48$, then $S_{10}$ is equal to

If ${a_1},\;{a_2},............,{a_n}$ are in $A.P.$ with common difference , $d$, then the sum of the following series is $\sin d(\cos {\rm{ec}}\,{a_1}.co{\rm{sec}}\,{a_2} + {\rm{cosec}}\,{a_2}.{\rm{cosec}}\,{a_3} + ...........$$ + {\rm{cosec}}\;{a_{n - 1}}{\rm{cosec}}\;{a_n})$

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

The sum of first $n$ natural numbers is

If $x,y,z$  are in $A.P.$ and  ${\tan ^{ - 1}}x,{\tan ^{ - 1}}y$ and ${\tan ^{ - 1}}z$ are also in other $A.P.$ then  . . .