Calculate

$$
S_{1}=1+2+3+4+\ldots . .+2014+2015+2016
$$

and

$$
S_{2}=2016+2015+2014+2013+\ldots .+3+2+1
$$
It is clear by commutation that

$$
S_{1}=S_{2}
$$

Furthermore,

$$
S_{1}+S_{2}=2017+2017+\ldots+2017=2017 \times 2016
$$

Therefore,

$$
S_{1}=\frac{2016 \times 2017}{2}
$$

We can generalize this calculation to any sequence with a constant progression:

Proposition 5. Let $\left(u_{n}\right)$ be a sequence such that for all ranks $\mathrm{n}$ integer: $u_{n+1}=u_{n}+b$ with $\mathrm{b}$ real. Then:

$$
u_{n}=u_{0}+n b
$$

and

$$
u_{0}+u_{1}+\ldots+u_{n}=\frac{(n+1)\left(u_{0}+u_{n}\right)}{2}
$$