Trouble understanding recursion in lines dividing a plane.

Question

In how many maximum regions does $n$ lines divide a plane into?

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It has a well-known recursive formula: $a_{n+1}=a_n+n+1$, where $a_n$ is the number of regions that $n$ lines divide a plane into. I am having a bit trouble understanding the logic behind it. I get that the $n+1^{\text{th}}$ line intersects the previous $n$ lines at $n$ points and everytime it intersects, it creates a new region, so we add $n$. I am having trouble understanding why we add $1$ at the end. What is that extra region?

Answer 1

Here is an image I provided to OEIS A000124 (the Lazy Caterer's sequence):

Consider adding the third cut (so going from $4$ regions to $7$). It splits

• the region above the intersection with the first cut
• the region between the intersections with the first and second cuts
• the region below the intersection with the second cut

so the third cut adds (up to) $3$ regions, and similarly with the earlier and later other cuts, demonstrating why $$a_{n+1}=a_n+(n+1).$$