Show that $n$ lines separate the plane into $\frac{n^2+n+2}{2}$ regions if no two of these lines are parallel and no three pass through a common point.
I know we start with the base case, where, if we call the above equation P(n), P(0), for 0 lines would be 0. But I really have no idea how to begin the inductive step. How do we know what k+1 we're supposed to arrive at?
Thanks!
Best Answer
Here is the way I usually think about this (it sort of uses induction).
With $0$ lines, there is $1$ region and no intersections of lines.
Each time a line is added and it crosses $k$ other lines it adds $k+1$ regions and $k$ intersections. Another way of looking at this is that for each line and $k$ intersections added, $k+1$ regions are added (the number of added lines and intersections).
Therefore, the number of regions is $1+\text{the number of lines}+\text{the number of intersections}$. With $n$ lines, there are $\binom{n}{2}$ intersections (if no two lines are parallel and no three lines are coincident).
Thus, the number of regions is $\binom{n}{2}+n+1=\frac{n(n-1)}{2}+n+1=\frac{n^2+n+2}{2}$.