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If \( S_{n}=\frac{3 \times 1^{3}}{1^{2}}+\frac{5 \times\left(1^{3}+2^{3}\right)}{1^{2}+2^{2}}+\frac{7 \times\left(1^{3}+2^{3}+3^{3}\right)}{1^{2}+2^{2}+3^{2}}+ \) Which of the following is correct?

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Sn = \(\frac{3\times1^3}{1^2}+\frac{5\times(2^3+2^3)}{1^2+2^2}+\frac{7\times(1^3+2^3+3^3)}{1^2+2^2+3^2}\) + .... + \(\frac{2n+1)(1^3+2^3+3^3+....+n^3)}{1^2+2^2+3^2+....+n^2}\)

∴ ΣTn = \(\cfrac{(2n+1)\frac{n^2(n+1)^2}4}{\frac{n(n+1)(2n+1)}6}\) 

\(=\frac32 n(n+1)\) 

\(=\frac32(n^2+n)\)

∴ Sn = ΣTn = \(\frac32\)(Σn2 + Σn)

 = \(\frac32(\frac{n(n+1)(2n+1)}6+\frac{n(n+1)}2)\) 

 = \(\frac34\)n(n + 1)\((\frac{2n+1}3+1)\) 

 = \(\frac14\)n(n + 1) (2n + 4)

 = \(\frac12\)n (n + 1) (n + 2)

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