Question Number 190061 by sonukgindia last updated on 26/Mar/23
Answered by a.lgnaoui last updated on 29/Mar/23
$$\mathrm{2}\left({n}^{{n}+\mathrm{1}} −{n}!\right)=\mathrm{5}{n}^{\mathrm{2}} −\mathrm{3}{n}−\mathrm{2} \\ $$$$\frac{\mathrm{2}\left({n}^{{n}+\mathrm{1}} \right)}{{n}!}−\mathrm{1}=\frac{{a}}{{n}!}\left({a}=\right) \\ $$$$\frac{\mathrm{2}{n}×{n}^{{n}} }{{n}!}=\frac{\mathrm{5}\left({n}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} −\frac{\mathrm{17}}{\mathrm{4}}}{{n}!}+\mathrm{1} \\ $$$$\frac{\mathrm{2}\left[\left({n}−\frac{\mathrm{3}}{\mathrm{2}}\right)+\frac{\mathrm{3}}{\mathrm{2}}\right]{n}^{{n}} }{{n}!}=\frac{\mathrm{5}\left({n}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} }{{n}!}−\left(\frac{\mathrm{13}}{\mathrm{4}{n}!}\right) \\ $$$$………….. \\ $$$$\left({a}\:{suivre}\right).. \\ $$