$\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\dots \dots \mathrm{n}$ terms =

$\begin{array}{rcl} & & \frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+.......\\ {\mathrm{t}}_{\mathrm{n}}& =& \frac{2\mathrm{n}+1}{\sum {\mathrm{n}}^2}\\ & =& \frac{{\displaystyle 2n+1}}{{\displaystyle \frac{\mathrm{n}(\mathrm{n}+1) (2\mathrm{n}+1)}{6}}}\\ {\mathrm{t}}_{\mathrm{n}}& =& \frac{6}{\mathrm{n}(\mathrm{n}+1)}=6\left[\frac{1}{\mathrm{n}}-\frac{1}{\mathrm{n}+1}\right]\\ {\mathrm{S}}_{\mathrm{n}}& =& \sum {\mathrm{t}}_{\mathrm{n}}\\ & =& 6\left[1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.........+\frac{1}{\mathrm{n}}-\frac{1}{\mathrm{n}+1}\right]\\ & =& 6\left[1-\frac{1}{\mathrm{n}+1}\right]\\ & =& \frac{6\mathrm{n}}{\mathrm{n}+1}\end{array}$