Problem 853

We denote $\mathcal{P}_{n}$ as the property « $\sum_{k=0}^{n} k=0+1+2+\cdots+n=\frac{n(n+1)}{2} »$.

- Initialization: $0=\frac{0 \times 1}{2}$, so $\mathcal{P}_{0}$ is verified.
- Heredity: Let $n$ be an integer such that $\mathcal{P}_{n}$ is verified, and let's show that $\mathcal{P}_{n+1}$ is also verified. $0+1+2+\cdots+n+(n+1)=\frac{n(n+1)}{2}+(n+1)$ by the induction hypothesis.

Therefore, $0+1+2+\cdots+n+(n+1)=(n+1)\left(\frac{n}{2}+1\right)=\frac{(n+1)(n+2)}{2}: \mathcal{P}_{n+1}$ is then verified.

By the principle of induction, we have then shown that $0+1+2+\cdots+n=\frac{n(n+1)}{2}$.