Sum of the first n squares

What the symbols mean

$S$

the sum $1^2+2^2+\cdots +n^2$

$n$

the last number squared

When to use it

Use it to count squares of every size on a grid (an 8 × 8 chessboard has $1^2+\cdots +8^2=204$ squares), to count cannonballs in a square pyramid, and when estimating areas under $y=x^2$ with rectangles, which is where integration starts.

Why it works

It can be proved by induction: assume it for n, add $(n+1{)}^2$, and the right side factorises into the same formula with $n+1$. One of n, $n+1$ is even and one of n, $n+1$, $2n+1$ is a multiple of 3, which is why the division by 6 always comes out exactly.

Common mistakes

• Squaring the sum instead: $(1+2+\cdots +10{)}^2=3025$, not 385.
• Writing $2n$ instead of $2n+1$ in the last bracket.