Trying to understand formula for counting regions of hyperplane arrangements in $\mathbb{R}^2$ There are up to $\binom{n}{2} + n + 1$ regions created by a hyperplane arrangement in $\mathbb{R}^2$ containing $n$ hyperplanes.
I want to understand this in a demonstrative way. Each hyperplane splits each region it passes in two regions. A newly added hyperplane therefore creates a new region each time it crosses another hyplerplane or the plane. So the number of regions should be $1$ (plane) $+ \binom{n}{2}$ (number of intersections of $n$ lines). But as I know the answer, I know that I am missing $n$ regions. 
Do you know where my mistake is? Why do I have to add the number of lines to get the max. number of regions? 

Suppose there are $p$ regions in the arrangement where you only consider the $n-1$ first lines. When adding the $n$-th line $\ell$, you keep all the old regions, and you split certain regions in two. To be precise, you add one extra region for every region in the hyperplan arrangement induced on $\ell$ by the first $n-1$ hyperplanes. This is a hyperplane arrangement in a one dimensional space, so it consists of a finite number ($\leq n-1$) of points in the line $\ell$. There are at most $n-1+1$ regions in this arrangement, so the number of regions in the $n$ line arrangement is at most $p+n$. If we let $M_n$ stand for the maximal number of regions in a plane hyperplane arrangement with $n$ hyperplanes, then $M_1=2$ and for all $n\geq 2$
$$M_{n}\leq M_{n-1}+n$$
Therefore $M_n\leq M_1+2+3+\cdots+n=1+\frac{n(n+1)}{2}=1+n+\binom{n}{2}$.
