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Sum Of Squares Calculator

Work out sum of squares instantly with clear inputs, formula shown and shareable results.

About Sum Of Squares Calculator

The sum of the first n squares has the closed form n(n + 1)(2n + 1)/6, which can be proved by induction or by telescoping the identity (k+1)³ − k³ = 3k² + 3k + 1.

Formula

Sum of squares

1² + 2² + … + n² = n(n + 1)(2n + 1) / 6

Worked example

With the default values below, Sum Of Squares Calculator returns Sum of the first n squares of 385. Change any input above and every figure here is recalculated in your browser.

Inputs used

How many terms naccepts 1–1,000,000
10
Also show the sum of this poweraccepts 1–6
4

Results

Sum of the first n squares
385
Sum of the first n values raised to your power
25,333
Mean square
38.500000
Largest square in the sum
100

Step by step

  1. 1Values used
  2. 2Sum of squares
  3. 3Sum of the first n squares= 385
  4. 4Sum of the first n values raised to your power= 25,333
  5. 5Mean square= 38.500000
  6. 6Largest square in the sum= 100

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is 1² + 2² + … + 10²?

10 × 11 × 21 / 6 = 385.

Is there a formula for any power?

Yes, Faulhaber's formula gives a polynomial of degree p+1 for each power p, with Bernoulli numbers as coefficients.

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