A while back, one of my friends challenged me to find out how many regions I can divide a plane into given $n$ lines. For instance:
He also told me that the formula to find the maximum number of lines on the plane is $$\frac{1}{2}\left(n^2+n+2\right).$$
How is this derived? (I would prefer an intuitive example as I have not studied any particularly advanced fields of geometry, set theory or whatever is needed to rigorously prove the formula).
Each new line you add can cut each of the other lines at most once. If you arrange it this way, then the line cuts the region that it belongs to before cutting the first line into two, and then every time it crosses a line, it cuts another region in two. So the number of new regions is the number of lines you had before plus 1. That is, the nth line adds n new regions. Taking into account that you start with 1 region when n = 0, you get $$ R = 1 + \sum^n_{i=1}i = 1 + \frac{n(n+1)}{2} = \frac{n^2 + n + 2}{2}. $$