\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}6Variables
- Define all variables, domains, and units before substitution.
- result: the quantity produced after the stated inputs and conditions are applied
How to use this formula
Computes the sum of the first n square numbers.
Important notes
- Verify assumptions and conventions before use.
Quick example
Apply the sum of the first squares to a small known case and verify the result.
Applicability, worked calculation, and verification
Domain and applicability
Apply Sum of the First Squares only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- Verify assumptions and conventions before use.
- For the Sum of the First Squares, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Sum of the First Squares, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- For the Sum of the First Squares, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Sum of the First Squares when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Sum of the First Squares, check zero, negative, and extreme input values before relying on the result.
- When using Sum of the First Squares, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}6 for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For n=5, 5·6·11/6=55.
- Compute n+1=6 and 2n+1=11.
- Multiply 5×6×11=330.
- Divide by 6 to obtain 55.
Independent verification
Direct addition 1+4+9+16+25 gives 55.
Common mistakes
- When copying Sum of the First Squares, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Sum of the First Squares, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Sum of the First Squares 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^2=\frac{n(n+1)(2n+1)}6. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sum of the First Squares.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Sum of the First Squares used for?
Computes the sum of the first n square numbers.
Can I copy this formula as LaTeX?
Yes. Copy \sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}6 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.