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

The sum of infinity terms of the series $\dfrac{1}{1+1^2+1^4} + \dfrac{1}{1+2^2+2^4} + \dfrac{3}{1+3^2+3^4}+....\infty$ is 

The sum of  $10$  terms of the series  $\left( x + \dfrac { 1 } { x } \right) ^ { 2 } + \left( x ^ { 2 } + \dfrac { 1 } { x ^ { 2 } } \right) ^ { 2 } + \left( x ^ { 3 } + \dfrac { 1 } { x ^ { 3 } } \right) ^ { 2 } + \ldots .$  is

Sum of the series $S=1^{2}-2^{2}+3^{2}-4^{2}+......-2008^{2}+2009^{2}$ is 

If $(1^{2}+2^{2}+3^{3}+.....12^{2})=650$, then the value of $(2^{2}+4^{2}+6^{2}+.......+24^{2})$ is 

the sum of the first n terms of the series ${ 1 }^{ 2 }+{ 2.2 }^{ 2 }+{ 3 }^{ 2 }+{ 2.4 }^{ 2 }+{ 5 }^{ 2 }+{ 2.6 }^{ 2 }....is\frac { n(n+1)^{ 2 } }{ 2 } $ when n is even.wheen n is odd the sum is

If the sum of first n positive integers is $ \frac{1}{5}  $ times the sum of their squares, then n equals

Let $\begin{bmatrix} n \ k\end{bmatrix}$ represents the combination of 'n' things taken 'k' at a time, then the value of the sum $\begin{bmatrix} 99\ 97\end{bmatrix}+\begin{bmatrix} 98\ 96\end{bmatrix}+\begin{bmatrix} 97\ 95\end{bmatrix}+...+\begin{bmatrix} 3\ 1\end{bmatrix}+\begin{bmatrix} 2\ 0\end{bmatrix}$ equals?

$\frac { 3 }{ 6 } +\frac { 3.5 }{ 6.9 } +\frac { 3.5.7 }{ 6.9.12 } +...\infty =$

Sum of the series
$P=\dfrac{1}{2\sqrt{1}+\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+....+\dfrac{1}{100\sqrt{99}+99\sqrt{100}}$ is

The sum of series $\sec^{-1}\sqrt {2}+\sec^{-1}\dfrac {\sqrt {10}}{3}+\sec^{-1}\dfrac {\sqrt {50}}{7}+...+\sec^{-1}\sqrt {\dfrac {(n^{2}+1)(n^{2}-2n+2)}{(n^{2}-n+1)^{2}}}$