Example 3 3 On the plane, there are $n$ lines, no two of which are parallel, and no three of which intersect at the same point. How many intersection points do these $n$ lines have in total?
Let $a_{n-1}$ be the number of intersection points that meet the requirements for $n-1$ lines. When adding the $n$-th line, this new line intersects each of the previous $n-1$ lines at exactly one point. Therefore, the number of intersection points that meet the requirements for $n$ lines is $a_{n}=a_{n-1}+(n-1), a_{2}=1$. Solving this gives $a_{n}=$ $\frac{n(n-1)}{2}$