Example 1 What is the maximum number of intersection points of $n$ lines on the same plane?
When $n=1$, there are no intersection points; when $n=2$, there is at most one intersection point; as $n$ increases, to ensure the maximum number of intersection points, each new line can be made to intersect with all the lines already on the plane. In this case, $n$ lines can have at most $0+1+\cdots+(n-1)=\frac{n(n-1)}{2}$ intersection points.

The above process can be represented as follows: Let $a_{n}$ denote the maximum number of intersection points of $n$ lines, then we have
$$
a_{1}=0, a_{n}=a_{n-1}+n-1 \text {. }
$$

Thus, $a_{n}=\frac{n(n-1)}{2}$.