1. Solve the system:

$$
\left\{\begin{array}{ccccccc} 
& x_{2}+ & x_{3}+ & \ldots & x_{n-1}+ & x_{n}= & 1 \\
x_{1}+ & & x_{3}+ & \ldots & x_{n-1}+ & x_{n}= & 2 \\
x_{1}+ & x_{2}+ & & \ldots & x_{n-1}+ & x_{n}= & 3 \\
\ldots & \ldots & \ldots & \ldots
\end{array}\right.
$$

where $n \in \mathbb{N}$.
Solution. If we add all the equations of the system, we get

$$
(n-1)\left(x_{1}+x_{2}+\ldots+x_{n-1}+x_{n}\right)=1+2+\ldots .+n
$$

i.e.

$$
x_{1}+x_{2}+\ldots+x_{n-1}+x_{n}=\frac{n(n+1)}{2(n-1)}
$$

If we subtract the $i$-th equation of the system from the last equation, we get

$$
x_{i}=\frac{n(n+1)}{2(n-1)}-i ; i=1,2, \ldots, n
$$