How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines have a common point. Further, this is not a non-Euclidean problem, but I wouldn't mind a discussion on the non-Euclidean nature of the problem.
I was thinking it might be easier to show using the unit sphere.
A GOOD REFERENCE I FOUND THAT WILL BE USEFUL TO ANYONE COMING ACROSS THIS PROBLEM IS Concrete Mathematics, Graham, Knuth, Patashnik pp. 4–8.
To prove it by induction, you first prove the base case: 0 lines divide the plane into 1 region.
Then you suppose that $n$ lines divide the plane into $\frac12(n^2+n+2)$ regions, and show that $n+1$ lines divide the plane into $\frac12((n+1)^2+(n+1)+2)$ regions. Since $\frac12((n+1)^2+(n+1)+2) - \frac12(n^2+n+2) = n+1$, you need to show that adding the $n+1$'th line adds exactly $n+1$ regions, or equivalently that the $n+1$th line can be made to pass through $n+1$ of the existing regions, dividing each one into two regions.
Does that help?