Sum of the first n whole numbers

What the symbols mean

$S$

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

$n$

the last (and largest) number added

When to use it

Use it for handshakes in a group (each of n people shakes hands with everyone else: $\frac{n(n-1)}{2}$, the sum up to $n-1$), the number of pins or cans in a triangular stack, or any running total of 1, 2, 3, … . Solving for n tells you whether a number is triangular.

Why it works

Write the sum twice, once forwards and once backwards, and add column by column: every column is $n+1$, and there are n columns, so twice the sum is $n(n+1)$. The picture shows the same idea with dots.

Common mistakes

• Forgetting to halve: $n(n+1)$ counts every number twice.
• Using it for sums that do not start at 1. For 51 + … + 100, subtract the sum up to 50 from the sum up to 100.