Number Sequences
Identifying the rule in a sequence and finding missing terms, including arithmetic and geometric patterns.
What is Number Sequences?
A number sequence is a list of numbers that follow a pattern or rule. In the 11+ exam, you need to identify the rule and use it to find the next term or a missing term in the sequence.
Common sequence types include arithmetic sequences (add or subtract the same amount each time), geometric sequences (multiply or divide by the same amount), square number sequences, and sequences with two-step rules.
Step-by-Step Method
Find the differences between consecutive terms
Write the difference between each pair of numbers. If the differences are all the same, it is an arithmetic sequence.
If differences are not constant, look at the differences of the differences
Second differences that are constant suggest a quadratic pattern (like square numbers).
Check for multiplication or division patterns
If each term is a multiple of the previous term, it is a geometric sequence (e.g. 2, 6, 18, 54 – multiply by 3).
Look for well-known sequences
Square numbers (1, 4, 9, 16…), cube numbers (1, 8, 27…), triangular numbers (1, 3, 6, 10…), Fibonacci-type (each term is the sum of the two before).
Check your rule works for ALL terms
Apply your rule to every term in the sequence. If it does not work for all of them, the rule is wrong.
Worked Examples
Find the next term: 3, 7, 11, 15, …
Working
- Differences: 7-3=4, 11-7=4, 15-11=4.
- The rule is “add 4 each time”.
- Next term: 15 + 4 = 19.
Find the missing term: 2, 6, 18, ___, 162
Working
- Check ratios: 6/2=3, 18/6=3.
- The rule is “multiply by 3 each time”.
- Missing term: 18 x 3 = 54.
- Check: 54 x 3 = 162. Correct!
What is the pattern in: 1, 4, 9, 16, 25, …?
Working
- 1 = 1 x 1, 4 = 2 x 2, 9 = 3 x 3, 16 = 4 x 4, 25 = 5 x 5.
- These are the square numbers.
- Next term: 6 x 6 = 36.
Common Mistakes
Assuming all sequences add the same amount, without checking for multiplication.
Always check whether the differences or the ratios are constant.
Only looking at the first difference and assuming the rule.
Check your rule works for EVERY term, not just the first two.
Confusing the position number with the term value.
The 5th term is not necessarily 5. Count carefully which position you need.
Top Tips
- Write the differences between terms below the sequence – it makes patterns much easier to spot.
- If the question gives you a position (e.g. “find the 10th term”), you may need to use the rule repeatedly or find a formula.
- Common two-step rules: “multiply by 2 then add 1” or “add an increasing amount” (1, 3, 6, 10 – adding 2, 3, 4…).
- If you cannot see the pattern, try looking at odd-positioned and even-positioned terms separately – some sequences alternate between two rules.
Ready to practise?
Put these techniques into action with our free practice papers.
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