Some sequences turn up so often they have names. They don't follow a single term-to-term rule like "add 3" — instead each one is built from a picture or a small trick. Three are worth recognising on sight, because once you spot the pattern you can keep going forever.

Triangular numbers

Stack dots into a triangle: one dot, then a row of two beneath it, then a row of three… Counting all the dots gives the triangular numbers:

Each one adds the next whole row, so the gaps grow by one each time: +2, then +3, then +4. This is one example of an nth-term rule that isn't linear — the step keeps changing.

Watch the triangle grow one row at a time. Step through it.

Square numbers

Now arrange the dots in a square instead — a 1\times1 square, then 2\times2, then 3\times3. Counting the dots gives the square numbers:

Each one is a whole number multiplied by itself — using index notation, the nth square number is n^2. So the 5th square number is 5^2 = 25.

The Fibonacci sequence

The Fibonacci sequence needs no picture — just a rule: start with two 1s, then each new term is the sum of the previous two.

Check it: 1+1=2, 1+2=3, 2+3=5, 3+5=8. To continue, just add the last two numbers you have.

See it explained

Khan Academy builds triangular numbers from dots, just like the diagram above.