For the series S =   $\frac{1}{(1+3)}{(1+2)}^2+\frac{1}{(1+3+5)}{(1+2+3)}^2+\frac{1}{(1+3+5+7)}{(1+2+3+4)}^2+\mathrm{....}$

$$\begin{gathered} S=1+\frac{1}{(1+3)}{(1+2)}^2+\frac{1}{(1+3+5)}{(1+2+3)}^{2.}+\mathrm{........} \\ {T}_r=\frac{1}{r^2}{(1+2+\mathrm{.....}+r)}^2=\frac{r^2+2r+1}{4} \\ {T}_7=16 \\ {S}_{10}=\sum _{r=1}^{10}{T}_r=\frac{1}{4}\{\frac{10\times 11\times 21}{6}+10\times 11+10\}=\frac{505}{4} \end{gathered}$$