3 days ago
- The article derives the formula for the sum of cubes (1³ + 2³ + ... + n³) using a difference table method, as an alternative to a graphical method shown in a Numberphile video.
- A table of values for n and the sum N is generated, and successive differences are computed until a constant fourth difference of 6 is found, indicating N is a fourth-degree polynomial.
- The leading coefficient (a₄) is determined as 1/4 from the constant difference, and then the remaining coefficients are found by solving simultaneous equations (using Python, Mathematica, or a spreadsheet) or by a stepwise recursive subtraction method.
- The stepwise method involves repeatedly subtracting the known polynomial term and analyzing the resulting table until all coefficients are zero, yielding the final formula N = n²(n+1)²/4.
- The author notes that while not as elegant as the graphical method, the difference table approach is systematic and requires no special ingenuity.