Let L(n) be the number of regions that n lines divide the plane (with
no two parallel and no three concurrent).
Clearly L(1) = 2.
Suppose we add lines one by one, and examine how the number of regions
When we add the nth line, it will intersect each of the preceding n-1
lines in a distinct point.
These intersection points divide the nth line into n segments.
Each such segment divides an existing region into two regions, creating
n new regions.
As a result: