Sum of the series $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\dots \dots \dots$ upto n terms is equal to

Given Series is $\frac{\mathit{3}}{\mathit{(}\mathit{1}\mathit{\times }\mathit{2}{\mathit{)}}^{\mathit{2}}}\mathit{+}\frac{\mathit{5}}{\mathit{(}\mathit{2}\mathit{\times }\mathit{3}{\mathit{)}}^{\mathit{2}}}\mathit{+}\frac{\mathit{7}}{\mathit{(}\mathit{3}\mathit{\times }\mathit{4}{\mathit{)}}^{\mathit{2}}}\mathit{+}\mathit{\dots }\mathit{\dots }$

$T_{\mathit{n}}\mathit{=}\frac{\mathit{3}\mathit{+}\mathit{(}\mathit{n}\mathit{-}\mathit{1}\mathit{)}\mathit{\times }\mathit{2}}{\mathit{\left\{}\mathit{n}\mathit{\right(}\mathit{n}\mathit{+}\mathit{1}\mathit{)}{\mathit{\}}}^{\mathit{2}}}\mathit{=}\frac{\mathit{(}\mathit{n}\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}\mathit{-}{\mathit{n}}^{\mathit{2}}}{{\mathit{n}}^{\mathit{2}}\mathit{(}\mathit{n}\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}}$

$T_{\mathit{n}}\mathit{=}\frac{\mathit{1}}{{\mathit{n}}^{\mathit{2}}}\mathit{-}\frac{\mathit{1}}{\mathit{(}\mathit{n}\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}}$

$\mathit{\therefore }\mathit{\text{Sum of series}}\mathit{=}\frac{\mathit{1}}{{\mathit{1}}^{\mathit{2}}}\mathit{-}\frac{\mathit{1}}{{\mathit{2}}^{\mathit{2}}}\mathit{+}\frac{\mathit{1}}{{\mathit{2}}^{\mathit{2}}}\mathit{-}\frac{\mathit{1}}{{\mathit{3}}^{\mathit{2}}}\mathit{\dots }\mathit{\dots }\mathit{.}\frac{\mathit{1}}{{\mathit{n}}^{\mathit{2}}}\mathit{-}\frac{\mathit{1}}{\mathit{(}\mathit{n}\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}}$

$\mathit{=}\mathit{1}\mathit{-}\frac{\mathit{1}}{\mathit{(}\mathit{n}\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}}\mathit{=}\frac{\mathit{n}\mathit{(}\mathit{n}\mathit{+}\mathit{2}\mathit{)}}{\mathit{(}\mathit{n}\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}}$