Regions cut by n lines

Idea. Let $R_n$ be the number of regions. Compare $R_{n+1}$ and $R_n$.

Induction. For $n=0$ there is one region, and $\frac{0+0+2}2=1$.

Suppose the formula holds for $n$ lines and add an $(n+1)$-th line $\ell$. It crosses each of the $n$ existing lines at $n$ distinct points. These $n$ points cut $\ell$ into $n+1$ pieces.

Each piece lies inside one existing region and splits it into two. Thus exactly $n+1$ new regions are created:
$$R_{n+1}=R_n+n+1.$$

Computation.
$$\frac{n^2+n+2}2+n+1=\frac{(n+1)^2+(n+1)+2}2,$$
which is the formula for $n+1$.

Remark. Equivalently $R_n=1+\binom n1+\binom n2$.

Lesson. To count regions or objects built incrementally, count the change produced by the last object added.