# Find the sum of the series upto n terms 1*3*5+3*5*7+5*7*9.............

$\begin{array}{l}1\cdot3\cdot5+3\cdot5\cdot7+5\cdot7\cdot9.....\ \\ n^{th}\ term\left(T_n\right)=\left(2n-1\right)\cdot\left(2n+1\right)\cdot\left(2n+3\right)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =8n^3+12n^2-2n-3\\ \text{Sum of n term}\left(S_n\right)=\Sigma\left(8n^3+12n^2-2n-3\right)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =8\cdot\Sigma n^3+12\cdot\Sigma n^2-2\cdot\Sigma n-3\cdot\Sigma1\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =8\cdot\left(\frac{n\left(n+1\right)}{2}\right)^2+12\left(\frac{n\left(n+1\right)\left(2n+1\right)}{6}\right)-2\frac{n\left(n+1\right)}{2}-3n\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =n\left(2n^3+8n^2+7n-2\right)\end{array}$

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