For example, continuing from the previous example, now we are asked how many non-intersecting regions $n$ lines can divide a plane into?

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Let the number of regions formed by $n-1$ lines be $a_{n-1}$, then $a_{1}=2, a_{2}=4$. When adding the $n$-th line, it is equivalent to adding $n-1$ points on the $n$-th line, dividing this line into $n$ segments (2 of which are rays), each segment divides the original region into two, thus adding $n$ regions. Therefore, the number of regions formed by $n$ lines is $a_{n}=a_{n-1}+n$, solving this gives $a_{n}=$ $\frac{n(n+1)}{2}+1$.