We trace $n$ lines in the plane, no two parallel, no three concurrent. Into how many parts is the plane divided?
When there are no lines, there is 1 sector, with one line there are 2 sectors. Let $u_{n}$ be the number of sectors when there are $n$ lines. When adding the $(n+1)$-th line, it will intersect the $n$ other lines, meaning it will pass through $n+1$ sectors. It will cut each of these sectors in two, thus adding $n+1$ regions. We obtain the following recurrence relation: $u_{n+1}=u_{n}+n+1$. It is easy to prove by induction that then

$$
u_{n}=\frac{n(n+1)}{2}+1
$$