In the sequence {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}........ of sets, the sum of elements in the ${\mathrm{n}}^{\mathrm{th}}$ set is

$\left\{1\right\}, \left\{2, 3\right\}, \left\{4, 5, 6\right\}, \left\{7, 8, 9, 10\right\}$

${\mathrm{S}}_{\mathrm{n}}$ last term $=\sum =\frac{\mathrm{n}(\mathrm{n}+1)}{2}$

first term $\mathrm{a}=\sum (\mathrm{n}-1)+1$

                    $=\frac{\mathrm{n}(\mathrm{n}-1)}{2}+1$

d=1,  no. of terms = n

$\begin{array}{rcl}{\mathrm{S}}_{\mathrm{n}}& =& \frac{\mathrm{n}}{2} \left[\mathrm{a}+1\right]\\ & =& \frac{\mathrm{n}}{2} \left[\frac{\mathrm{n}(\mathrm{n}-1)}{2}+1+\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right]\\ & =& \frac{\mathrm{n}}{2} \left(\frac{2{\mathrm{n}}^2}{2}+1\right)=\frac{\mathrm{n}\left({\mathrm{n}}^2+1\right)}{2}\end{array}$