A square of side n (n natural) is divided into n2 squares of side 1. Each pair of "horizontal" boundary lines and each pair of "vertical" boundary lines enclose a rectangle (a square is also considered a rectangle). A rectangle has a length and a width; the width is less than or equal to the length.
(a) Prove that there are 8 rectangles of width n−1.
(b) Determine the number of rectangles with width n−k (0≤k≤n−1,k integer).
(c) Determine a formula for 13+23+...+n3. geometryrectanglecombinatoricscombinatorial geometry