Induction on geometric configurations

Into how many parts do $n$ lines in general position (i.e., no two of which are parallel and no three of which meet at a point) divide the plane?

Prove that $n$ distinct lines in the plane divide it into regions that can be colored with white and black paint so that adjacent regions are colored with different colors.

Prove that $n$ distinct lines in the plane divide it into regions, which can be colored white and black so that adjacent regions are colored with different colors.

Prove that $n$ distinct lines in the plane, no two of which are parallel and no three of which meet at one point, divide the plane into $\left(n^{2}+n+2\right) / 2$ regions.

Let in a convex $n$-gon, $n \geqslant 4$, no three diagonals meet at one point. Prove that it has $D(n)=\frac{(n-3)(n-2)(n-1) n}{24}$ points of intersection of diagonals.

Let $n \geqslant 1$ spheres in space be arranged so that any three of them have exactly two common points, and for any fourth sphere $S$ one of these points lies inside $S$ and the other outside $S$. Prove that space is divided by these spheres into $N(n)=2 n+\frac{n(n-1)(n-2)}{3}$ pieces.