Tag: forming an arithmetic progression between two quantities a and b

The average of certain first consecutive even number is $101$. Find their sum?

The sum of the series
$ _{  }^{ 4n }{ { C } _{ 0 } }+ _{  }^{ 4n }{ { C } _{ 4 } }+ _{  }^{ 4n }{ { C } _{ 8 } }+........ _{  }^{ 4n }{ { C } _{ 4n } }$ is 

The sum of the series $\overset { n }{ \underset { r=0 }{ \sum   }  } (r^2+1)(r!)$ is

$1+6+9(\dfrac{1^2 +2^2 +3^2}{7}) +12(\dfrac{1^2 +2^2 +3^2+4^2}{9} )+15(\dfrac{1^2 +2^2 +3^2+4^2+5^2}{11}) +$_____
Find sum of $15$ terms

Find the sum of 1 + $\dfrac{1}{4} + \dfrac{1.3}{4.8} + \dfrac{1.3.5}{4.8.12} +.......\infty $

Sum of values of  $x ,$  which we should substitute in  $( 1 )$  to give the sum of the series :  $C _ { 0 } + C _ { 4 } + C _ { 8 } + C _ { 12 } + \ldots \ldots ,$  is -

If  $0 < x , y , a , b < 1 ,$  then the sum of the infinite terms of the series
 $\sqrt { x } ( \sqrt { a } + \sqrt { x } ) + \sqrt { x } ( \sqrt { a b } + \sqrt { x y } ) + \sqrt { x } ( b \sqrt { a } + y \sqrt { x } ) + \ldots$  is

If  $S _ { n }$  denotes the sum of the terms in the  $n ^ { t h }$  bracket of the series $( 1 ) + ( 3 + 5 ) + ( 7 + 9 + 11 ) + ( 13 + 15 + 17 + 19 ) + \ldots \ldots , \text { then } \left( S _ { 11 } - S _ { 9 } \right) =$

The sum of the infinite terms of the series $\cot^{-1}\left(1^{2}+\dfrac{3}{4}\right)+\cot^{-1}\left(2^{2}+\dfrac{3}{4}\right)+\cot^{-1}\left(3^{2}+\dfrac{3}{4}\right)+..$ is equal to:

Sum infinite terms of the series $\cot ^ { - 1 } \left( 1 ^ { 2 } + \frac { 3 } { 4 } \right) + \cot ^ { - 1 } \left( 2 ^ { 2 } + \frac { 3 } { 4 } \right) + \cot ^ { - 1 } \left( 3 ^ { 2 } + \frac { 3 } { 4 } \right) + \ldots$ is