Example 1. On a plane, there are ? straight lines, any two of which intersect, and no three lines intersect at the same point. How many regions will these lines divide the plane into?
Solution: Obviously, the number of intersection points of $n$ lines is $p_{n}=C_{n}^{2}$, so the number of regions they divide the plane into is
$$
S_{n}=C_{n}^{2}+n+1=\frac{1}{2}\left(n^{2}+n+2\right) .
$$