Number Sequences: Find the Rule

A8Sequences

Spot the rule, continue the pattern.

Sequences follow a term-to-term rule — find it, then extend or fill gaps.

What comes next?

3, 7, 11, 15, ?

+4 +4 +4 → rule: add 4

next term: 15 + 4 = 19

Check the rule works for EVERY gap before trusting it.

Learn the 3-Step Method

1Find the gaps.

7 − 3 = 4, 11 − 7 = 4, 15 − 11 = 4 — the difference each time.

gaps of 4

2State the rule.

‘Add 4’ (sequences can also subtract, or even double).

rule: + 4

3Extend or fill gaps.

Next: 15 + 4 = 19. A missing middle term: use the rule from its neighbour.

19, 23, 27…

💡 Key IdeaSequences can go DOWN (20, 17, 14 — subtract 3) or multiply (2, 4, 8, 16 — double). Always test the gaps, don't assume adding.

Worked Example

Fill the gap: 40, 34, ?, 22, 16.

Find the rule

34 − 40 = −6 · 16 − 22 = −6

rule: subtract 6

→

Apply from a neighbour

34 − 6 = 28

→

✓ Check

28 − 6 = 22 ✓

Watch Out! Common Mistakes

Testing only the first gap.

1, 2, 4, 8 → “add 1” ✕

Check every gap! 2→4 adds 2, 4→8 adds 4 — this sequence doubles. Next: 16.

Going the wrong direction.

20, 17, 14 → next 17 ✕

It’s decreasing! The rule is subtract 3: next is 11.

Filling a gap from the wrong side.

Careful! Use the neighbour BEFORE the gap with the forward rule (34 − 6), or the one after with the reverse (22 + 6) — both must agree.

⬇ Cheat Sheet

SEQUENCES

Find the gap between EVERY pair of terms.
State the rule (add, subtract, double…).
Apply the rule to extend or fill gaps.
Check a filled gap against both neighbours.

Examples: 3, 7, 11 → +4 → 15, 19 · 40, 34, ?, 22 → 28.

📖 Continue Your Journey

✓ A1 What Is Algebra?✓ A2 Function Machines ✓ A3 Writing Expressions ✓ A4 Substitution ✓ A5 Simplifying Expressions ✓ A6 One-Step Equations ✓ A7 Two-Step Equations A8 Sequences A9 The nth Term A10 Formulae A11 Bar Models for Unknowns A12 Algebra Problem Solving

→

Next Lesson: A9 The nth TermJump straight to the 100th term.

📄 Free printable: Download the Sequences cheat sheet (PDF) — keep it by your child's desk.

Quick summary

Find the difference between every pair of neighbouring terms.
State the term-to-term rule: add, subtract or multiply.
Test the rule on every gap — 1, 2, 4, 8 doubles rather than adds.
Fill gaps from either neighbour and check both directions agree.

Frequently asked questions

What is a term-to-term rule?

The instruction that gets you from each term to the next, like 'add 4' for 3, 7, 11, 15 or 'double' for 2, 4, 8, 16.

How do you find a missing middle term?

Work the rule forward from the term before it (34 − 6 = 28) and check backward from the term after (22 + 6 = 28) — they must match.

What sequence types appear in the 11+?

Adding and subtracting rules, doubling, and two-step patterns — plus the nth-term shortcut covered in our next lesson.

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A7 Two-Step Equations