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Number sequences

A sequence is a list of numbers that follows a rule. Find the rule by looking at the gaps, and you can find the next number, a missing number or even the 100th number.

  • Ages 9 to 11
  • 11+ Maths
  • Worked examples

What is it?

A number sequence is a list of numbers (called terms) that follow a pattern. The first number is the 1st term, the next is the 2nd term, and so on.

Most 11+ sequences add or take away the same amount each time. Others multiply, have gaps that grow, use square numbers, or add the two numbers before. Harder questions ask for a rule for any term, called the nth term.

A typical question

What is the next number in this sequence? 6, 11, 16, 21, …

How to spot it

  • A list of numbers with '…' or a blank box at the end or in the middle.
  • 'What is the next term?', 'Find the missing number' or 'What is the 20th term?'.
  • Patterns of shapes (matchsticks, dots or tiles) where each picture gets bigger: count them and you have a sequence.
  • 'Write the rule' or 'Find the nth term' in harder papers.

Step-by-step method

  1. Step 1: Write the difference (the gap) between each pair of numbers above the sequence, with + or −.
  2. Step 2: If every gap is the same, keep adding (or taking away) that amount.
  3. Step 3: If the gaps change, look at the gaps themselves: do they go up by 1, 2 or the same amount each time?
  4. Step 4: If the numbers grow very fast, try multiplying: is each term ×2, ×3 or ×10 the one before? If they shrink fast, try ÷.
  5. Step 5: Still stuck? Check for square numbers (1, 4, 9, 16…), triangle numbers (1, 3, 6, 10…) or adding the two terms before.
  6. Step 6: Check your rule works for every term in the list, not just the first two, before you write the answer.

Worked examples

Example 1: the same gap each time

Question: Find the next number: 6, 11, 16, 21, …
  1. Step 1: Gaps: 6 → 11 is +5, 11 → 16 is +5, 16 → 21 is +5.
  2. Step 2: Every gap is +5, so the rule is 'add 5'.
  3. Step 3: 21 + 5 = 26.

Answer: 26

Example 2: a missing number in the middle

Question: Find the missing number: 31, ?, 19, 13
  1. Step 1: Use the two numbers next to each other that you know: 19 → 13 is −6.
  2. Step 2: Check: from 31 to 19 is two jumps of −6 (31 − 6 = 25, 25 − 6 = 19). ✓
  3. Step 3: So the missing number is 25.

Answer: 25

Example 3: gaps that grow

Question: Find the next number: 1, 2, 4, 7, 11, …
  1. Step 1: Gaps: +1, +2, +3, +4. The gaps are not the same.
  2. Step 2: The gaps go up by 1 each time, so the next gap is +5.
  3. Step 3: 11 + 5 = 16.

Answer: 16

Example 4: multiplying

Question: Find the next number: 2, 6, 18, 54, …
  1. Step 1: Gaps: +4, +12, +36. These grow too fast for an adding pattern.
  2. Step 2: Try multiplying: 2 × 3 = 6, 6 × 3 = 18, 18 × 3 = 54. The rule is '×3'.
  3. Step 3: 54 × 3 = 162.

Answer: 162

Example 5: the nth term (any term)

Question: The sequence 4, 7, 10, 13, … continues. Write the rule for the nth term, then find the 20th term.
  1. Step 1: The gap is +3 every time, so the rule starts with 3n (the 3 times table: 3, 6, 9, 12…).
  2. Step 2: Compare: the sequence is 4, 7, 10, 13 and the 3 times table is 3, 6, 9, 12. Each term is 1 more.
  3. Step 3: So the nth term is 3n + 1. Check the 2nd term: 3 × 2 + 1 = 7. ✓
  4. Step 4: 20th term: 3 × 20 + 1 = 61.

Answer: nth term = 3n + 1; the 20th term is 61

Common traps

  • Checking only the first gap

    In 1, 2, 4, 7, 11 the first gap is +1, but the next gaps are +2, +3, +4. Work out every gap before you decide on the rule.

  • Mixing up + and −

    In a sequence that goes down, like 50, 43, 36, the gap is −7. Write the sign above each gap so you keep going the right way.

  • Counting along to a far-away term

    Counting to the 100th term one step at a time is slow and easy to get wrong. Use the nth term rule instead.

  • Forgetting the extra bit in the nth term

    For 4, 7, 10, 13 the answer isn't just 3n. Compare with the times table and add or take away the difference (3n + 1).

  • Going below zero

    Sequences can carry on past zero: 5, 2, −1, −4 … Use a number line if negative numbers feel tricky.

Top tips

  • Always write the gaps above the numbers. It takes seconds and shows the pattern.
  • Know your special sequences: square numbers (1, 4, 9, 16, 25), triangle numbers (1, 3, 6, 10, 15), and the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13), where you add the two terms before.
  • For a missing number, use two terms that sit next to each other to find the gap.
  • In a shape pattern, count the shapes in each picture and you have an ordinary number sequence.
  • Check your answer by carrying the rule on one more step.

Practice questions

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11 practice questions on this topic, from easy to 11+ level. Reading the guide is free for everyone; practice is for members, and signing up is free.

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