Question

Suppose that l lines are drawn through a circle and these lines form p points of intersection (involving exactly two lines at each intersection). how many regions r are formed inside the circle by these lines?

Answer 1

When there are no lines, there is one region: (l, r) = (0, 1).

When there is one line, there are two regions: (l, r) = (1, 2).

When there are two lines, there are 4 regiions: (l, r) = (2, 4).

When there are 3 lines, there are 7 regions: (l, r) = (3, 7).

Each line adds a number of regions equal to its line number, so the total number of regions is one more than the sum of numbers 1 to l.

... r = (l)(l+1)/2 + 1