Dividing a plane

Let's say we have a plane. Draw N straight lines on the plane, any way you wish. Try to divide the plane into as many different regions as possible. How many regions is that? For example, if we draw 1 line on the plane, we can divide it into two regions. If we draw 2 lines, we can divide it into four regions.

Followup questions: Draw N perfect circles on a plane, of any size, anywhere you want. Into how many regions can you divide the plane? Next, draw N perfect ellipses on another plane. Into how many regions can you divide the plane?

This is, by the way, what's called a combinatorics puzzle. Combinatorics is the branch of mathematics which is all about counting. Fabricated statistics say that 50% of my readers will run away at the first mention of math, but surely no one's afraid of a little counting?

Solution has been posted