Sum of the First n Squares Calculator

How it is calculated

The tool adds 1² + 2² + … + n² with the closed form S = n × (n + 1) × (2n + 1) / 6, so no term-by-term addition is needed. n is how many squares you add, counting from 1: the last term is n² itself. It must be a whole number of at least 1; zero, negatives and decimals give no result. The answer is an exact integer shown with thousands separators. The arithmetic is done in whole numbers, dividing by 6 before the final multiplication, so every digit is exact. The largest n accepted is 3,000,000, whose sum is 9,000,004,500,000,500,000; anything larger would pass the 64-bit integer limit and shows no result rather than a rounded guess.

Common mistakes

• Squaring the plain sum instead. (1 + 2 + … + n)² is the sum of cubes, not the sum of squares.
• Reading the answer as the sum of cubes. The cube sum is n² × (n + 1)² / 4, which is much larger.
• Forgetting the 2n + 1 factor, or using 2n − 1, which gives a different series.
• Expecting the series to start somewhere other than 1². To sum squares from 5² to 10², subtract the sum to 4 from the sum to 10.
• Entering a decimal such as 10.5 or a value above 3,000,000. Only whole n up to that limit gives a result.