Sequences
A row of numbers with one missing: at the end, in the middle, or far along the row. Here’s how the gaps between the numbers show you the rule every time, and the traps the wrong answers set.
Chapter 1
What are number sequence questions?
A sequence is a row of numbers that follows a rule. One number is missing, and you find it. The gaps between the numbers nearly always show you the rule.
The row. Usually four to six numbers. They go up, go down, or zigzag.
The question mark. Usually the next number, sometimes one in the middle. Some questions ask for the 12th or 20th number instead.
41, 43, 46, 50, ?
The options
- Option 1: 60
- Option 2: 55
- Option 3: 61
- Option 4: 54
A Maths question from Rush, level 2.
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The gap
How far it is from one number to the next. Write + if the row goes up and − if it goes down.
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The term
Each number in the row is a term: the 1st term, the 2nd term, and so on. Its place matters when you’re asked for the 12th.
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The rule
What takes you from each term to the next. It must fit every pair: 2, 4, 8, 14 looks like doubling, but 8 doubled is 16. The gaps grow by 2, so next is 22.
Chapter 2
Solve it in 3 steps.
Here’s the question, one step at a time.
The question
41, 43, 46, 50, ?
The options
- Option 1: 60
- Option 2: 55
- Option 3: 61
- Option 4: 54
A Maths question from Rush, level 2.
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Step 1
Write the gaps.
Under each pair of neighbours, write how far it is from one to the next. 41 to 43 is +2, 43 to 46 is +3, and 46 to 50 is +4. The gaps aren’t the same, so there’s no single step to add.
The gaps: +2, +3, +4.
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Step 2
Write the gaps between the gaps.
Do the same again, on the gaps. +2 to +3 is 1 more, and +3 to +4 is 1 more. The gaps grow by 1 each time, so the next gap is +5.
Each gap is 1 more, so the next is +5.
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Step 3
Add it on, then check every gap.
50 + 5 = 55. Now test the rule on the whole row: 41, 43, 46, 50, 55 goes up 2, 3, 4, then 5. It fits every gap, so it’s right.
55 fits: the gaps go 2, 3, 4, 5.
Three mistakes to spot
| The mistake | Do this instead |
|---|---|
| Using the last gap again. 54 is 50 + 4. But the gaps grow, so the next one isn’t 4. | Look at the gaps between the gaps: each is 1 more, so +5. |
| Adding the new gap twice. 60 is two lots of the new gap: 50 + 5 + 5. | One number on means one gap: 50 + 5. |
| Going one number too far. 61 is the number after 55: 55 + 6. | The question mark comes straight after 50. Stop at the first new number. |
So it’s 55. The gaps grow by 1 each time: +2, +3, +4, then +5.
Remember: Gaps, then gaps of gaps.
Chapter 3
The other rows Rush asks.
Each card is one kind of row: what to look for, one worked example, and the trap. Start with the gaps every time.
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A missing number in the middle
Find the step from two neighbours you can see. Then step on from the number just before the question mark.
252, 244, ?, 228, 220: 228 to 220 is −8, so the missing number is 244 − 8 = 236. Check the other side: 236 − 8 = 228. Trap: the answers can be only 1 or 2 apart, so a slip picks the wrong one.
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Gaps that shrink
Gaps can get smaller each time, as well as bigger.
50, 42, 35, 29, ?: the gaps go −8, −7, −6, so the next is −5, and 29 − 5 = 24. Trap: 23, from taking 6 again, or 34, from adding the 5 instead of taking it away.
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Zigzag: two steps take turns
Up, down, up, down? Two steps are taking turns.
15, 18, 16, 19, 17, 20, ?: the gaps go +3, −2, +3, −2, +3. The last was +3, so next is −2: 20 − 2 = 18. Trap: 23, from adding 3 again. A zigzag is two rows woven together too: 15, 16, 17 and 18, 19, 20.
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Two rows woven together
Numbers jumping about? Read every other number.
8, 46, 11, 40, 14, 34, ?: one row goes 8, 11, 14 (up 3) and the other 46, 40, 34 (down 6). The 7th number is in the first row: 14 + 3 = 17. Trap: 28, the next number in the wrong row.
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Times and halving
Gaps that double, or grow very fast? Try × 2 or × 3. Going down, try ÷ 2.
160, 80, ?, 20, 10: each number is half the one before, so the missing number is 80 ÷ 2 = 40. Check: 40 ÷ 2 = 20. Trap: 50, halfway between 80 and 20, which only works when the step stays the same.
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Crossing zero
Count back to 0 first, then on past it.
7, 3, ?, −5, −9 goes down 4 each time. From 3 it’s 3 back to 0, and 1 more: −1. Check: −1 − 4 = −5. Trap: 1, from dropping the minus sign.
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Decimal and fraction steps
Count in tenths (or eighths), then write it back.
0.8, 1.1, 1.4, 1.7, ? goes up 0.3. 7 tenths and 3 tenths make 10 tenths, a whole one, so 1.7 + 0.3 = 2. Trap: 1.10, from writing the 10 after the point. 3/8, 5/8, 7/8 goes up 2/8 the same way, so next is 9/8: one whole and an eighth.
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The 12th or 20th number
Count the steps, not the numbers: the 12th is 11 steps after the 1st.
2, 6, 10, 14, … goes up 4. The 12th is 11 steps of 4 after the 2: 2 + 11 × 4 = 2 + 44 = 46. Trap: 50, from taking 12 steps. Like a fence, 12 posts have 11 gaps.
Know these rows by heart
Spot one and you can skip the working. Their gaps give them away.
- Square numbersGaps 3, 5, 7, 9: the odd numbers.149162536496481100
- Triangle numbersGaps 2, 3, 4, 5: one more each time.13610152128364555
- Cube numbers1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3…182764125
- Add the two beforeEach gap is a number already in the row.11235813213455
Chapter 4: In the exam
Picture patterns.
The exam also draws sequences as pictures, asks for the first number, or asks for the rule itself. Rush doesn’t ask these, but the gaps still do the work. In verbal reasoning, the same number rows are called number series.
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Step 1
Count each pattern.
Pattern 1 has 4 sticks, pattern 2 has 7 and pattern 3 has 10. Now it’s a number sequence.
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Step 2
Find the gap.
Each pattern adds 3 sticks, not 4, because the new square shares a side with the one before. So the gap is +3.
A new square: 3 more sticks.
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Step 3
Count the steps.
Pattern 10 is 9 steps after pattern 1. So it’s 4 + 9 × 3 = 4 + 27 = 31.
- 4 + 9 × 3
- = 4 + 27
- = 31
Three mistakes to spot
| The mistake | Do this instead |
|---|---|
| Counting the steps wrong. 34 takes 10 steps of 3, and 28 takes 8. | Pattern 10 is 9 steps after pattern 1. |
| Four sticks for every square. That gives 40: ten squares of 4 sticks each. But each new square borrows a side. | Count the sticks in the pictures. Don’t guess the gap. |
| Leaving out the start. 30 is 10 × 3. It misses the extra stick the first square has. | Start from pattern 1’s 4 sticks, then add the steps. |
So it’s 31. Pattern 10 is 9 steps of 3 after pattern 1. Or: 3 sticks a square, plus 1 to start, is 3 × 10 + 1 = 31.
Remember: Count the pictures, then the gaps.
Two more ways it’s asked
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The first number
Work backwards: undo the step.
?, 17, 24, 31 goes up 7. Going back, take 7 away: 17 − 7 = 10. Trap: adding 7 going backwards. That gives 24, already in the row.
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n 1 2 3 4 3n 3 6 9 12 3n + 2 5 8 11 14 The rule for any term
Multiply the place (n) by the step, then fix the start.
5, 8, 11, 14 goes up 3, so try 3 × n. That makes the 1st term 3, but it’s 5, so add 2: the rule is 3n + 2. The 20th is 3 × 20 + 2 = 62. Trap: n + 3. The step times n, so it goes in front: 3n.
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Write the gaps.
- Write the gaps between the gaps.
- Add it on, then check every gap.
Remember: Gaps, then gaps of gaps.
For parents.
What parents ask us about sequences.
Questions parents ask
What is the nth term?
It's a rule that gives any number in a sequence from its position. For 5, 8, 11, 14 it is 3n + 2: the step (3) times the position, plus whatever makes the first one right. The 20th is 3 × 20 + 2 = 62. Rush asks it as "What is the 20th number in this sequence?", which you can also answer without the rule: 19 steps of 3 after 5 is 5 + 57 = 62.
Are number sequences in maths or verbal reasoning?
Both. Maths papers ask for the next or missing number. In verbal reasoning they're called number series, and two rules taking turns is common. The method is the same: write the gaps.
Which special sequences are worth knowing?
Square numbers (1, 4, 9, 16, 25), cube numbers (1, 8, 27, 64, 125), triangular numbers (1, 3, 6, 10, 15, 21) and the add-the-two-before pattern (1, 1, 2, 3, 5, 8). Recognising one saves working out every gap.
Rush: 11+ Practice & KS2 SATsPractise sequences in Rush.
- Sequences as 100-second games, easy at first and harder as your child improves
- Every miss kept with the right answer and why
- Sequences is in Premium, and six other games are free
On the App Store for iPhone and iPad
Six other games are free, plus a new family game every day. Sequences is in Premium: all 41 games are £24.99 a year after a free week1, or £3.99 a month.