16. Five marks exercise problems II

If \(S_1\), \(S_2\), \(S_3\), …, \(S_m\) are the sums of \(n\) terms of \(m\) A.P.'s whose first terms are \(1, 2, 3, … m\) and whose common differences are \(1, 3, 5, …, (2m - 1)\) respectively, then show that $S_1+S_2+S_3+...+S_m=\frac{1}{2}\mathit{mn}\left(\mathit{mn}+1\right)$.