Sum of cubes via difference tables

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] 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. 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…