prove that the greatest number of regions that $n \geq 1$ circles can divide the plane is $n^2-n+2$

Question

This is an induction problem, but I have no idea how to do something like this. Any hints?

Answer 1

Hint: Suppose that the result holds for $n$ and consider an arrangement with $n + 1$ circles. Pick a circle $C$. In at most how many points in total can $C$ meet the other $n$ circles?

Answer 2

Hint: Assume that you already have $k$ circles, and draw one more.

1. At how many points will that new circle intersect the earlier ones - at maximum?
2. How many arcs is that new circle split into (by those points of intersection)?
3. How many pre-existing regions are split in two (at maximum)?