I have no trouble solving for the base case. I need help solving the inductive step. I know that the nth line creates n new regions. But I don't know if that's based on intuition or if I have to prove that. Do I have to prove that the nth line creates n new regions? If so, how can it be done?
Induction is not needed. Any such configuration with $n$ chords is a plane polyhedral graph. Let $v$ be the number of vertices, $e$ the number of edges, $f$ the number of faces. Since each chord has two end points on the circle and each pair of chords determines a point of intersection, we have $$ v=2n+{n\choose 2}=\frac{n^2+3n}{2}.$$ More precisesly, there are $v_3=2n$ vertices of degree $3$ (on the circle) and $v_4={n\choose 2}$ vertices of degree $4$ (the intersections between chords). By the handshaking lemma we have $2e=3v_3+4v_4 $ so that $$e=3n+n(n-1)=n^2+2n. $$ Then from Euler's $v+f=e+2$ we obtain $$ f-1=\frac{n^2+n+2}{2}$$ (we need to subtract one because o fthe unbounded region).