Chapter 5.7, Problem 52ES

A single line divides a plane into two regions. Two lines (by crossing) can divide a plane into four regions, three lines can divide it into seven regions (see the figure). Let $P_n$ be the maximum number of regions into which n lines divide a plane, where n is a positive integer

a.Derive a recurrence relation for $P_k$ in terms of $P_{k-1},$ for each integer $k\ge 2.$
b. Use iteration to guess an explicit formula for $P_n.$