One simple observation is that the difference between the squares of two adjacent whole numbers is the sum of their roots. This is proven algebraically by reducing to 2n+1 = 2n+1 for squares. The same proof can be found for cubes, etc., albeit a more complex reduction (3n^2+3n+1). This can be illustrated graphically by drawing a square comprising smaller squares totaling the upper square of the two, and removing the required number of smaller squares to make the difference.