LaTeX
\sum_{k=1}^{n}k^3=\left(\frac{n(n+1)}{2}\right)^2Variables
- n: positive integer upper limit
- k: summation index
How to use this formula
Calculates the sum of the cubes from one through n.
Important notes
- The result is the square of the triangular number.
- The identity is useful in induction and finite-difference arguments.
Quick example
For n = 3, 1³ + 2³ + 3³ = 36.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The result is the square of the triangular number.
- For the Sum of the First n Cubes, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Sum of the First n Cubes, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- Confirm the finite-set, indexing, ordering, and counting conventions assumed by the expression.
Worked example
For n = 3, 1³ + 2³ + 3³ = 36.
Common mistakes
- When copying Sum of the First n Cubes, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Verify the result of Sum of the First n Cubes with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Sum of the First n Cubes in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\sum_{k=1}^{n}k^3=\left(\frac{n(n+1)}{2}\right)^2. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Sum of the First n Cubes used for?
Calculates the sum of the cubes from one through n.
Can I copy this formula as LaTeX?
Yes. Copy \sum_{k=1}^{n}k^3=\left(\frac{n(n+1)}{2}\right)^2 or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.