Suppose there are N lines in a 2-D space, and you have to maximize the no. of region in which the 2-D space get divided due to the these lines, lets Ln denote the maximum number of region created by n lines.

Is there any closed formula for Ln. ?

E.g, L0 = 1, L1 = 2, L2 = 4

How will the answer change for 3-D space, space being divided into regions by planes rather the lines ?

How will the answer change for N-D space, space being divided into regions by appropriate hyperplanes ?