Patterns
Find the rule, then use it
Number patterns are puzzles with rules. Once you find the rule, you can predict any term — and find what's missing.
See It in Action
The Problem
What is the next number in the sequence: 3, 7, 13, 21, 31, ___?
Common Mistake
The pattern goes up by 4 each time. 31 + 4 = 35.
Check: 7 − 3 = 4, 13 − 7 = 6. The differences are not the same, so it's not a simple "add 4" rule. The student only checked the first pair.
Correct Approach
Find the differences: 7−3=4, 13−7=6, 21−13=8, 31−21=10. The differences increase by 2 each time (4, 6, 8, 10…). So the next difference is 12. Next term: 31 + 12 = 43.
Answer: 43
Differences: 4, 6, 8, 10, 12. Each goes up by 2. The sequence: 3, 7, 13, 21, 31, 43. ✓
The Core Concept
A number pattern (or sequence) is a list of numbers where each term follows the same rule. The most common types are: arithmetic (add or subtract the same amount each time: 3, 7, 11, 15…), geometric (multiply or divide by the same amount: 2, 6, 18, 54…), and two-step rules (the rule itself changes, like +2, +4, +6, +8…).
The key skill is finding the rule from the pattern, not just from one pair of terms. Always check the rule on at least two pairs before trusting it. If a sequence is 2, 4, 8, 16 — the difference between terms grows (2, 4, 8) so this isn't an arithmetic sequence. It's geometric: multiply by 2 each time.
Once you have the rule, you can work forwards (find the next term) or backwards (find a missing term earlier in the sequence). For working backwards, reverse the operation: if the rule is "add 5," to find a previous term you "subtract 5."
Common Mistakes to Avoid
Only checking one pair of differences
Always check at least two gaps. A two-step or growing pattern won't be caught by checking only the first pair.
Confusing arithmetic and geometric patterns
In 3, 6, 12, 24 — each term doubles (×2), not adds 3. If the differences grow (3, 6, 12) it's likely geometric. If they stay the same (3, 3, 3) it's arithmetic.
Not finding the rule before applying it
Always state the rule clearly ("add 7 each time") before finding the missing term. Guessing without a stated rule leads to errors.
Forgetting that missing terms can be in the middle, not just at the end
If given 5, ___, 17, 23 — you need to work out the rule from the known terms and fill in the gap (rule: add 6, so missing term = 11).
Try It Yourself
Find the missing term: 80, 40, ___, 10, 5.
Hint: Look at what happens between each term. Is something being added, subtracted, multiplied, or divided?
Key Tips
Always write the differences between terms below the sequence before guessing the rule.
If differences are constant → arithmetic. If differences grow → two-step or geometric.
State the rule before applying it: "The rule is ×3 each time, so the next term is..."
Ready to practise Patterns?
Reading is the start. The Maths Gym has exercises designed around this skill — with instant feedback and progress tracking.