Question Number 10290 by konen last updated on 02/Feb/17
$$\mathrm{3}×\mathrm{3}!+\mathrm{4}×\mathrm{4}!+...+\mathrm{12}×\mathrm{12}!=? \\ $$
Answered by ridwan balatif last updated on 02/Feb/17
$$\mathrm{3}×\mathrm{3}!+\mathrm{4}×\mathrm{4}!+...+\mathrm{12}×\mathrm{12}! \\ $$$$\left(\mathrm{4}−\mathrm{1}\right)×\mathrm{3}!+\left(\mathrm{5}−\mathrm{1}\right)×\mathrm{4}!+\left(\mathrm{6}−\mathrm{1}\right)×\mathrm{5}!+...+\left(\mathrm{13}−\mathrm{1}\right)×\mathrm{12}! \\ $$$$\mathrm{4}×\mathrm{3}!−\mathrm{1}×\mathrm{3}!+\mathrm{5}×\mathrm{4}!−\mathrm{1}×\mathrm{4}!+\mathrm{6}×\mathrm{5}!−\mathrm{1}×\mathrm{5}!+...+\mathrm{13}×\mathrm{12}!−\mathrm{1}×\mathrm{12}! \\ $$$$\mathrm{4}!−\mathrm{3}!+\mathrm{5}!−\mathrm{4}!+\mathrm{6}!−\mathrm{5}!+...+\mathrm{13}!−\mathrm{12}! \\ $$$$−\mathrm{3}!+\mathrm{13}! \\ $$$$\mathrm{13}!−\mathrm{3}! \\ $$