Maths is the science of patterns.

Extend the pattern — and watch the rule appear.

Here's a growing pattern made of dots. Each new term (one step of the pattern) is a little block of 2 rows. Press Add the next term and keep going. Watch two things at once: the dots stacking up, and the numbers underneath. Somewhere around the third or fourth term, you'll notice the rule reveal itself.

Stop and feel how big that is. You didn't just guess the next number — you found a machine. Tell the machine a position (say, 50) and it hands you that term instantly: 2 × 50 = 100. You never had to count 50 blocks of dots. That's the whole point of a rule: it lets you skip ahead to any term you want, even ones too far away to draw.

And notice why the rule has to keep working forever, not just for the terms you drew. Every block is built the same way — two rows, with as many dots in each row as the term's position. There's no special term where that suddenly changes. Because the building method is the same every time, the rule 2n must hold every time. That's the difference between a lucky guess and a real generalisation: you can point to the reason it can't break.

Two kinds of rule — and why one is far stronger.

There are two ways to describe 2, 4, 6, 8… and the difference is the most important idea on this page. A term-to-term rule tells you how to get from one term to the next one: here, “add 2.” A position-to-position rule (also called the nth term) tells you a term straight from its position: here, “double the position,” 2n. Pick a far-off position below and race the two rules to it.

This is exactly why generalising matters. “Add 2” keeps you walking the sequence one step at a time, forever tied to the term before. The rule 2n cuts the rope: every term stands on its own position, ready whenever you ask. When teachers say “find the nth term,” this is what they're really asking for — the rule that doesn't need the previous answer.

How to find the nth term, in three looks.

Most patterns you'll meet in school are linear — they go up (or down) by the same amount each step, like a staircase with even steps. Every linear rule has the same shape: an + b, which just means “multiply the position by some number a, then add some number b.” Here's the friendly way to crack one, no algebra required.

Look 1 — find the gap. See how much the sequence jumps each step. For 5, 8, 11, 14 the gap is 3. That gap is your a, so the rule starts with 3n.

Look 2 — test the bare rule. Try 3n on its own: for n = 1 it gives 3, but your sequence starts at 5. So 3n is always 2 short.

Look 3 — adjust. Add whatever fixes the gap: here, +2. The rule is 3n + 2. Check it: position 1 → 3 + 2 = 5. Position 4 → 12 + 2 = 14. It works for every term you can see, so you trust it for the ones you can't.

The gap tells you what to multiply by. A quick check tells you what to add. That's nearly every linear pattern, cracked.

The recipe works going down too. Take 20, 17, 14, 11. The gap is −3 (you subtract 3 each step), so the rule contains −3n. Test it: at n = 1, −3n gives −3, but the sequence starts at 20 — that's 23 too low. So add 23, and the rule is −3n + 23. Check position 4: −12 + 23 = 11. A downhill staircase is just an uphill one with a negative gap; the method never changes.

One gentle warning that we'll come back to: the gap is not the whole rule. A gap of 3 means the rule contains 3n — but you almost always need the little “+ b” adjustment too. Forgetting it is the single most common slip, so the next demo lets you feel both pieces moving.

Build the rule yourself.

Pick a sequence, then dial in your rule with the two sliders: the multiplier a and the adjuster b. Each cell turns green when your rule's term matches the real one, and red when it doesn't. Get all five green and you've found the nth term.

Play with a few sequences and a habit forms. The gap fixes the multiplier; the start fixes the adjuster. “Even numbers” give 2n, “odd numbers” give 2n − 1, the three-times table gives 3n. Same recipe, different dials — which is exactly what makes a single method feel powerful.

Patterns you can build out of shapes.

Patterns aren't only numbers — they grow as shapes too, and the same rule-finding works. Here's a row of squares made from matchsticks, where neighbouring squares share a side. Press Add a square and watch the matchstick count. Can you feel where the rule comes from before the numbers tell you?

Notice what just happened: the algebra came from the picture. The “3n” is the three repeated sides; the “+ 1” is the one-off starter. When you can point to where each piece of a rule lives in the shape, you're not memorising a formula — you're understanding it. That's the difference generalising makes.

Here's something lovely: there's more than one correct way to see the same rule. You could instead count 4 sticks for the first square, then 3 for each extra one — that's 4 + 3(n − 1), which tidies up to exactly 3n + 1. Different eyes, different picture, same rule. Two people can describe a pattern in words that look nothing alike and still be saying the same true thing — and checking that they match is itself a neat little taste of algebra.

Spot the rule.

Time to trust your eye. For each sequence below, pick the rule — the nth term — that makes it. Use the recipe: find the gap, then check position 1. There's an explanation after every answer, so a wrong guess is just a faster way to learn.

When the steps aren't even.

So far every pattern has climbed by the same gap each step — a straight staircase. But some patterns speed up as they go, and the very same habit (look, extend, name the rule) still cracks them. You just have to notice that the gaps themselves form a pattern.

Take the square numbers: 1, 4, 9, 16, 25. The gaps are 3, 5, 7, 9 — not steady, but growing by 2 each time. The clue is in the name: each one is a number times itself (1×1, 2×2, 3×3, 4×4), so the nth term is n × n, written n². You can even see it — term 3 really is a 3-by-3 square of dots. Picture beats formula again.

Or the triangular numbers: 1, 3, 6, 10, 15 — the dots you'd use to build bigger and bigger triangles (or stack bowling pins). Each term adds one more row than the last, so the gaps go 2, 3, 4, 5. These don't follow a simple gap-times-n rule, and that's fine: the lesson isn't “every rule is an + b.” The lesson is that a pattern always has a rule, and your job is to hunt for it — sometimes in the numbers, sometimes in the gaps, sometimes in a picture. Linear staircases are simply the friendliest place to start.

If the terms don't climb evenly, look at the gaps. A pattern in the gaps is still a pattern.

Three traps when you generalise.

Pattern-spotting feels easy, which is exactly why it's easy to slip. Here are the three mix-ups that catch nearly everyone — and how to dodge them.

Once you see rules, you can't unsee them.

Generalising isn't a school trick — it's how we describe a world that loves to repeat. Each card below is a real pattern with its own nth term quietly running underneath.

In every one of these, the win is the same: instead of listing forever, you capture the whole thing in one short rule. That's why algebra — letters standing for “any position” — is so useful. It's the language we invented precisely so we could say “for every n” in a single breath.

And this is only the beginning of the story. The moment you write a rule like 2n or 3n + 1, that little n stops being a position number and becomes a variable — a letter that can stand for anything. Later you'll learn to combine these rules, rearrange them, and solve for the n that makes something true (“which seat row holds exactly 50 people?”). Every bit of that grows from the single move you practised today: looking at a few examples and daring to write down the rule behind all of them.