From pattern to rule: square and triangular numbers

A pattern becomes powerful when you can write a rule for it. A rule lets you jump straight to step 100 without drawing 100 steps.

Linear patterns

If a pattern grows by the same amount each time, it is linear. Tiles: 4, 7, 10, 13… grow by 3 each step, so the rule is t = 3n + 1. Graphed, it makes a straight line.

Square numbers

1, 4, 9, 16, 25… are square numbers: n × n, written n². You can draw each one as a square of dots. The differences between them are 3, 5, 7, 9: the odd numbers.

Triangular numbers

1, 3, 6, 10, 15… are triangular numbers. Two copies of the same triangle fit together to make a rectangle n by n + 1, so the triangle is half of that: n(n + 1) ÷ 2. That picture is a proof: it shows why the rule works for every n, not just the ones you checked.

Testing and proving

Checking a few cases builds confidence. A proof explains why the rule must always hold. Mathematicians want both.