Problem of the Week, #117

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

PROBLEM 117

What is the largest number of regions $r(n)$ that a plane is divided into by $n$ straight lines in the plane?
Give $r(n)$ as a function of $n$ and explain why your answer is correct.

PROBLEM 117A

What is the largest number of regions $r(n, d)$ that $d$-dimensional Euclidean space is divided into by $n$ hyperplanes?
Give $r(n, d)$ as a function of $n$ and $d$, and find formulas for $b(n, d)$, the number of regions that are bounded, and for $u(n, d)$, the number of regions that are unbounded.
Of course, $r(n, d) = b(n, d) + u(n, d).$ Explain why these numbers are correct.