Sum of Cubes Calculator

About this calculator

The sum of cubes 1³ + 2³ + 3³ + ... + n³ has a remarkably clean closed form: it always equals [n(n+1)/2]² — in other words, the sum of the first n cubes is exactly the square of the sum of the first n integers. So for n = 4, the sum 1+8+27+64 = 100 equals (1+2+3+4)² = 10², and that pattern holds for every n.

This identity is a staple of algebra and discrete-math courses as an example of proof by mathematical induction, and it’s one of the most quoted “beautiful” identities in elementary number theory precisely because it links a cubic sum to a squared linear sum with no obvious reason why they should match. Beyond the classroom, the closed form lets programmers and engineers compute large cube sums in constant time instead of looping through every term, which matters in algorithm-complexity analysis and in generating test cases for numerical software.