# Which of the following is correct?

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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)