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

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

\(=\frac32 n(n+1)\)

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

∴ S_{n} = ΣT_{n} = \(\frac32\)(Σn^{2} + 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)