Question Number 62878 by mathmax by abdo last updated on 26/Jun/19
$${calculate}\:{min}\:\sum_{\mathrm{0}\leqslant{i}\leqslant{n}\:{and}\:\mathrm{0}\leqslant{j}\leqslant{n}} \:\:\:\:{i}.{j} \\ $$
Answered by mr W last updated on 26/Jun/19
$$\sum_{\mathrm{0}\leqslant{i}\leqslant{n}\:{and}\:\mathrm{0}\leqslant{j}\leqslant{n}} \:\:\:\:{i}.{j} \\ $$$$=\sum_{\mathrm{0}\leqslant{i}\leqslant{n}} \sum_{\mathrm{0}\leqslant{j}\leqslant{n}} \:\:\:\:{i}.{j} \\ $$$$=\sum_{\mathrm{0}\leqslant{i}\leqslant{n}} {i}.\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}.\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$=\frac{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{4}} \\ $$
Commented by mathmax by abdo last updated on 26/Jun/19
$${thanks}\:{sir}\:{mrw}. \\ $$