Chapter E, Problem 40E

Prove formula (e) of Theorem 3 using the following method published by Abu Bekr Mohammed ibn Alhusain Alkar-chi in about ad 1010. The figure shows a square ABCD in which sides AB and AD have been divided into segments of lengths 1, 2, 3, …, n. Thus the side of the square has length n(n + 1)/2 so the area is [n(n + 1)/2]². But the area is also the sum of the areas of the n “gnomons” G₁, G₂, ..., G_n shown in the figure. Show that the area of G_i is i³ and conclude that formula (e) is true.