$\text{The sum to n-terms of the series 2,5,10,17,26,… is}$

${\mathrm{S}}_{\mathrm{n}}=2+5+10+17+\dots +{\mathrm{T}}_{\mathrm{n}-1}+{\mathrm{T}}_{\mathrm{n}}-------\left(1\right) \left(differences of the terms 3,5,7,... are in A.P\right)$

$\begin{array}{l}{\mathrm{S}}_{\mathrm{n}}= 2+5+10+.......+{\mathrm{T}}_{\mathrm{n}-1}+{\mathrm{T}}_{\mathrm{n}}----\left(2\right)\\ \left(1\right)-\left(2\right)\Rightarrow 0=2+3+5+7+\dots +(\mathrm{n}-1)\mathrm{term}]-{\mathrm{T}}_{\mathrm{n}}\\ {\mathrm{T}}_{\mathrm{n}}=2+\frac{n-1}{2}\left(2\left(3\right)+\left(n-2\right)2\right)\\ {\mathrm{S}}_{\mathrm{n}}=\sum t_n=\sum \left(n^2+1\right)=\frac{n\left(n+1\right)\left(2n+1\right)}{6}+n\\ =\frac{n\left(2n^2+3n+7\right)}{6}\end{array}$