Sum of Cubes via Difference Tables

Summary

Next post Previous post July 19, 2026 at 9:07 PM by Dr. Drang Yesterday I watched the most recent Numberphile video, in which Ben Sparks explains a few finite series and the equations that simplify the calculation of their sums. He derives the formulas graphically, and it’s all very cleverly done, especially the one for the sum of cubes. VIDEO The idea is to get a simple polynomial expression in n for N=∑m=1nm3 As I said, Ben’s graphical method is really clever and easy to understand, and it leads to this expression: N=14n2(n+1)2 I wanted to see if I could get that same result using a nongraphical and less clever method. The method that came to mind was to generate a handful of values, put them in a table, and start taking differences. Here are some values of n, N, and the first four differences: n N Δ Δ² Δ³ Δ⁴ 1 1 8 19 18 6 2 9 27 37 24 6 3 36 64 61 30 4 100 125 91 5 225 216 6 441 The differences are calculated by looking in the preceding column and subtracting the value in the same row from the value in the following row. This sort of thing is fairly easy to do by hand but is really easy to do in a spreadsheet. Here’s a screenshot of the table in Numbers, where I’m displaying the formula for the first difference, Δ: That formula can be filled into all the difference cells, which is why it’s so easy. (I included only the first six rows in the table above because I figured you’d trust me that all the values in the fourth difference column, Δ⁴, are the same.) The constant values in the fourth difference column mean N is a fourth-degree polynomial in n: N=a0+a1n+a2n2+a3n3+a4n4 Because the constant fourth difference is 6, we know that a4=64!=624=14 This comes from the fact that differences are analogous to derivatives, and d4Ndn4=4⋅3⋅2⋅1a4= 24a4 Once we know the degree of the polynomial and have a4, we can figure out the other coefficients by solving a set of four simultaneous equations for four different values of n and N. For example: a0+a1⋅1+a2⋅12+a3⋅13+14⋅14=1 a0+a...