How many regions do n straight lines divide the plane, if no two lines
are parallel, and no three lines are concurrent (intersect at a single
point)?
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
changes.
When we add the nth line, it will intersect each of the preceding n1
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:
L(n) 
= 
n + L(n1) 

= 
n + (n1) + (n2) + ... + L(1) 

= 
n + (n1) + (n2) + ... + 2 

= 
n(n+1)/2 + 1 

= 
(n^{2} + n + 2)/2 