What is the formula for the sum of cubes from 1 to n?

The closed-form is [n(n+1)/2]², which equals the square of the n-th triangular number. For example, 1³+2³+3³+4³ = [4×5/2]² = 10² = 100.

Does this tool handle negative numbers?

Yes. In "Custom list" mode you can enter negatives. The cube of a negative number is negative (e.g. (-3)³ = -27), so the sum reflects that.

What is the largest n I can use?

JavaScript integers are safe up to 2⁵³-1. For very large n the result may exceed that and lose precision; the tool warns you when n > 100,000.

Why is the sum of cubes equal to the square of the triangular number?

This identity was known to Nicomachus (~100 AD). A visual proof arranges unit cubes into an L-shaped gnomon pattern showing that each successive layer contributes exactly k³ units, and the total area is a square of side 1+2+…+n.

Can I use decimal numbers in custom list mode?

Yes. Decimals are fully supported; each value is cubed (raised to the power 3) and the results are summed. The output is rounded to 10 significant figures to avoid floating-point noise.

Sum of Cubes Calculator

Name: Sum of Cubes Calculator
Availability: InStock
Author: Nham Vu

Find the sum of cubes from 1 to n using the closed-form formula, or enter a custom list of numbers to cube and add together.

Mode

Enter n

Sum = 1³ + 2³ + … + n³

Enter a value and click Calculate to see the result.

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Summary

Find the sum of cubes from 1 to n using the closed-form formula, or enter a custom list of numbers to cube and add together.

How it works

Choose "1 to n" mode and enter a positive integer n, or switch to "Custom list" mode.
In "1 to n" mode the calculator applies the closed-form identity: 1³+2³+…+n³ = [n(n+1)/2]².
In "Custom list" mode enter any numbers separated by commas or spaces; each is cubed then summed.
Results display the final sum, the expanded formula, and a step-by-step breakdown.
Copy the result or the full expansion with one click.

Use cases

Verifying closed-form cube-sum proofs in number theory coursework.
Checking series summation answers in calculus or discrete math.
Computing weighted totals where each term is raised to the third power.
Exploring the relationship between triangular numbers and cube sums.
Quick sanity-check for competitive programming solutions involving cubic series.
Teaching the identity 1³+2³+…+n³ = (1+2+…+n)² interactively.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu