janmr blog

The sum of the first $n$ squares is

$$s_n = \sum_{k=1}^n k^2 = \textstyle\frac{1}{6} n (n+1) (2n+1) \quad .$$

The numbers $s_0, s_1, s_2, \ldots$ are called the square pyramidal numbers.

Many different proofs exist. Seven different proofs can be found in Concrete Mathematics and even a visual proof has been published (via @MathUpdate).

One of the simplest proofs uses induction on n. This approach assumes that you know (or guess) the correct formula beforehand, though.

This post will show a derivation which is a formalization of the derivation shown on wikipedia. It revolves around manipulating sums and the fact that

$$k^2 = \sum_{j=1}^k (2j-1)$$

since $k^2 - (k-1)^2 = 2k-1$.

We will now write $s_n$ in three different ways. The first simply inserts the above expression for $k^2$:

$$s_n = \sum_{k=1}^n \sum_{j=1}^k (2j-1) \quad .$$

The second reverses the order of summation for the inner sum:

$$s_n = \sum_{k=1}^n \sum_{j=1}^k (2(k-j)+1) \quad .$$

The third starts as the first and does a series of manipulations:

$$\begin{aligned} s_n &= \sum_{k=1}^n \sum_{j=1}^k (2j-1) = \sum_{j=1}^n \sum_{k=j}^n (2j-1) = \sum_{j'=1}^n \sum_{k=n+1-j'}^n (2(n+1-j')-1) \\ &= \sum_{j'=1}^n \sum_{k'=1}^{j'} (2(n-j')+1) = \sum_{k=1}^n \sum_{j=1}^k (2(n-k)+1) \end{aligned}$$

(the manipulations being: Switching the order of summation, change of variable $j' = n+1-j$, change of variable $k' = k+j'-n$, renaming $j' \rightarrow k$, $k' \rightarrow j$).

We now add together these three expressions for $s_n$ and get

$$3 s_n = \sum_{k=1}^n \sum_{j=1}^k (2n+1) = (2n+1) \sum_{k=1}^n k = (2n+1) \frac{n (n+1)}{2}$$

which, after dividing each side by 3, produces the wanted formula.