Question Number 212991 by efronzo1 last updated on 28/Oct/24
$$\:\:\:\:\:\underbrace{\downharpoonleft\underline{}\:} \\ $$
Answered by Frix last updated on 28/Oct/24
![H(n)=Σ_(k=1) ^n k!(k^2 +k+1) ((H(n)+1)/((n+1)!))=n+1 [test it for n=1, 2, 3,...] ⇒ answer is 2026](https://www.tinkutara.com/question/Q213018.png)
$${H}\left({n}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:{k}!\left({k}^{\mathrm{2}} +{k}+\mathrm{1}\right) \\ $$$$\frac{{H}\left({n}\right)+\mathrm{1}}{\left({n}+\mathrm{1}\right)!}={n}+\mathrm{1}\:\:\:\:\:\left[\mathrm{test}\:\mathrm{it}\:\mathrm{for}\:{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},…\right] \\ $$$$\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\mathrm{2026} \\ $$