Number Sequences & Patterns Worksheets

Find the next term, then say why

Each problem shows the opening terms and a question mark. Two things are wanted: the next number, and the rule that produced it.

The second half is what makes this worth setting. A child can extend 3, 6, 9, 12 by adding without ever articulating that the rule is add three, and until they can articulate it they cannot use it on anything else. The answer key gives both, so the rule is markable rather than assumed.

Three kinds of pattern, in the order they get taught

Arithmetic adds or subtracts a constant. 4, 7, 10, 13. Subtracting versions appear too, so a child meets a descending sequence rather than assuming patterns only go up.

Geometric multiplies or divides by a constant. 2, 6, 18, 54. Halving sequences appear as well, which is the case that catches out anyone who has decided geometric means getting bigger.

Quadratic is where the second differences are constant rather than the first. These are kept simple, generally of the form n² or n² plus a constant, so the rule stays findable by a Key Stage 3 pupil rather than requiring the full nth-term machinery.

Mixed rotates all three, which is revision rather than teaching.

Why patterns are the bridge to algebra

Spotting that a sequence goes up by four every time is arithmetic. Writing that as a rule that predicts any term is algebra, and the gap between those two is the whole of early secondary maths.

Which is why the rule matters more than the next term. A pupil who can only produce the next number is doing repeated addition; one who can say add four each time has generalised, and generalising is the skill that transfers.

Quadratic sequences make that concrete in a way nothing else does at this age, because you physically cannot extend one by guessing. You have to find the second difference, and finding it is the method.

How many terms to show

Three to seven, five by default, and it changes the difficulty more than the pattern type does.

Three terms is genuinely ambiguous for anything beyond arithmetic: 2, 4, 8 could be doubling or it could be n² minus something, and a bright pupil who spots the second reading is not wrong. Five terms removes almost all of that. Seven makes even a quadratic sequence obvious once the second differences are lined up.

So use fewer terms to make a sheet harder, not a different pattern type. That is the lever most people miss.

Two things to watch

Geometric sequences grow fast. Seven terms of a sequence tripling each time reaches four figures quickly, which is fine as a stretch and less good for a Year 5 sheet where the arithmetic then swamps the pattern-spotting. Shorten the sequence rather than avoiding geometric altogether.

And ask for the rule out loud before it is written. Most pupils can see a pattern several seconds before they can describe it, and the describing is the part that needs practice.