Question Number 197320 by tri26112004 last updated on 13/Sep/23
$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}\left({k}+\mathrm{1}\right)} \\ $$
Answered by Erico last updated on 13/Sep/23
![∀k∈[1,n] (−1)^(k(k+1)) =1 ⇒ Σ_(k=1) ^n (−1)^(k(k+1)) =n](https://www.tinkutara.com/question/Q197324.png)
$$\forall\mathrm{k}\in\left[\mathrm{1},\mathrm{n}\right]\:\:\:\:\:\left(−\mathrm{1}\right)^{\mathrm{k}\left(\mathrm{k}+\mathrm{1}\right)} =\mathrm{1} \\ $$$$\Rightarrow\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{k}\left(\mathrm{k}+\mathrm{1}\right)} =\mathrm{n} \\ $$