What is the maximum number $L_n$ of regions defined by $n$ lines in the plane? ### Solution When a line passes through $m$ number of regions, it adds $m$ to the number of total regions Every $n^{\text{th}}$ line drawn can intersect at most $(n-1)$ lines, passing through $n$ regions - $L_0=1$ is the number regions in an undivided plane - $L_1=2$ after splitting the plane once - $L_2=4$ since this line split both regions - $L_3=7$ since the new line can intersect two lines, it passes though three regions. Those three regions are doubled, and the untouched regions are included $L_{n}=L_{n-1}+n$ which is the triangle numbers added to the initial region $L_n=1+\sum_{k=0}^n{k}=1+\frac{n(n+1)}{2}$ to validate the $n+1$ case from [[Proof by Induction]] $L_{n+1}=\frac{(n+1)(n+2)}{2}=\frac{n(n+1)+2(n+1)}{2}=\frac{n(n+1)}{2}+n+1=L_n+n \quad \square$