# Sum of squares of first n natural numbers

Level: A Level Maths

The sum of the squares of the first n natural numbers is defined as:

${\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}}$

It can be written using sigma notation as:

${\displaystyle \sum _{r\mathop {=} 1}^{n}r^{2}}$

## Formula

Given ${\displaystyle n}$, there is a formula for calculating the sum:

${\displaystyle \sum _{r\mathop {=} 1}^{n}r^{2}={\frac {n}{6}}(n+1)(2n+1)}$

## Proof

The formula can be proved by mathematical induction.