Official solution
\[{T_r} = {{(2r + 1)({1^3} + {2^3} + {3^3} + ...... + {r^3})} \over {{1^2} + {2^2} + {3^2} + ...... + {r^2}}}\]
\[{T_r} = (2r + 1){\left( {{{r(r + 1)} \over 2}} \right)^2} \times {6 \over {r(r + 1)(2r + 1)}}\]
\[{T_r} = {{3r(r + 1)} \over 2}\]
Now, \[S = \sum\limits_{r = 1}^{10} {{T_r}} = {3 \over 2}\sum\limits_{r = 1}^{10} {({r^2} + r)} \]
\[ \Rightarrow {3 \over 2}\left\{ {{{10 \times (10 + 1)(2 \times 10 + 1)} \over 6} + {{10 \times 11} \over 2}} \right\}\]
\[ \Rightarrow {3 \over 2}\left\{ {{{10 \times 11 \times 21} \over 6} + 5 \times 11} \right\}\]
\[ \Rightarrow {3 \over 2} \times 5 \times 11 \times 8 = 660\]