Number Patterns: Why the Next Number Is Easy and the 50th Is Not

10 October 2026 · by Larry

He can carry on 5, 8, 11, 14 without a pause. 17, 20, 23. Ask him for the next number and he is right every time, a bit pleased with himself.

Then the question asks for the 50th number in the pattern, and he sits there. Or he writes 155, confidently, and loses the mark. The correct answer is 152.

This is one of the quietest gaps in primary maths, because the easy half looks like understanding. He has noticed that the numbers go up by 3. That is a real observation. But it is a way of continuing a pattern, and the question has stopped asking him to continue it. It is asking him to jump.

Why the 50th is a different kind of question

Continuing a pattern needs one fact: what gets added each time. Finding a distant term needs a second fact that nobody told him he needed: where the pattern starts, and how many jumps it has taken to get there.

Take 5, 8, 11, 14. The 50th number is not 50 lots of 3 plus 5. It sounds right, and that is why it is such a popular wrong answer. Work it through: 50 × 3 = 150, plus 5 is 155.

But the first number, 5, is already there before any jump happens. To reach the 2nd number there is one jump. To reach the 3rd there are two. To reach the 50th there are only 49 jumps. So it is 5 + 49 × 3 = 5 + 147 = 152.

The slip is the classic one in this topic, an off-by-one: counting the numbers when he should have been counting the gaps between them. It is the same mistake as counting fence posts and fence panels, and children make it for the same reason, because both feel like the same count.

The test that settles it in ten seconds

Before any rule, ask him to check his method on a term he can already see. Take the 3rd number, which is 11.

• If his method is “3 × 3 + 5”, he gets 14. But the 3rd number is 11. The method is out by one jump.
• If his method is “5 + 2 × 3”, he gets 11. It matches.

This is the habit worth teaching: never trust a rule until it gives you back a number you already know. It costs ten seconds, it works on any pattern, and it catches the off-by-one before the paper is marked instead of after.

Two ways to find the rule, and why to teach the first one first

The first is the one above, built from the jumps: start number, plus (term number minus 1) lots of the step. It is slower to say, but every part of it can be pointed to in the list of numbers, which is why a child can rebuild it himself under exam pressure.

The second is the shortcut. For 5, 8, 11, 14 the rule can be written as “3 times the term number, plus 2”, because 3 × 1 + 2 = 5, 3 × 2 + 2 = 8, and so on. For the 50th: 3 × 50 + 2 = 152, the same answer.

The shortcut is faster, but a child who learns it as a formula to memorise has no way to know when it is wrong. A child who has built it from the jumps can see where the “+ 2” came from: it is the start number, 5, with one step of 3 taken back off. Teach the jump version first. Let the shortcut turn up later, once he can explain it.

Shape patterns: where the same slip hides in a picture

School papers often give these as matchstick or square patterns rather than lists of numbers. One square uses 4 matchsticks. Two squares in a row use 7. Three in a row use 10.

Children count the sticks in the picture and then guess. Ask him instead: what do I add each time a square joins the row? Three. The first square costs 4 because it has to stand on its own, and every square after it costs only 3, because it shares a side with the one before.

So 20 squares in a row use 4 + 19 × 3 = 61 matchsticks. The 19 is the number of squares added after the first, the same jumps-not-numbers idea, now in a drawing. Check it on the picture first: three squares should give 4 + 2 × 3 = 10, and it does.

The most useful question a parent can ask here is “which part of the picture is new each time, and which part was always there?” The always-there part is the start. The new part is the step.

Patterns that are not “add the same each time”

Not every pattern adds a constant, and children who have only seen those get stuck quickly. Three other kinds are worth recognising:

• Doubling or multiplying. 3, 6, 12, 24. The gaps are 3, 6, 12, so they are not constant. Each number is twice the one before. The tell is that the differences are themselves growing.
• Growing gaps. 1, 2, 4, 7, 11. The gaps are 1, 2, 3, 4. The pattern is in the gaps, not in the numbers.
• Two patterns taking turns. 2, 5, 4, 7, 6, 9, 8, 11. Read the 1st, 3rd, 5th, 7th numbers on their own and you get 2, 4, 6, 8. Read the 2nd, 4th, 6th, 8th and you get 5, 7, 9, 11. Two simple patterns interleaved look like one confusing one. Carrying on, the 9th number is the 5th of the first strand, which is 10.

The habit that covers all three is the same, and a child can do it with a pencil: write the gaps between the numbers underneath, and look at the gaps before looking at the numbers. Constant gaps mean add. Growing gaps mean look at the gaps. Gaps that double mean multiply. If the gaps look like a mess, try every other number.

A question that tests whether he really has it

Ask: is 100 in the list 5, 8, 11, 14, and so on?

A child who only knows how to continue the list will start writing numbers and probably give up around 60. A child who has understood it notices that every number in the list is 2 more than a number in the 3 times table (5 is 3 + 2, 8 is 6 + 2, 11 is 9 + 2). Is 100 two more than a multiple of 3? 98 is not in the 3 times table, so no. What about 101? 99 is, so yes, and 101 is the 33rd number.

He does not need to be able to do this at P4. But if he can, the topic is secure, and it also quietly connects to factors and multiples and to how the times tables are stored, because both are doing the work underneath.

The five-minute check tonight

• “What is the 10th number in 5, 8, 11, 14?” The answer is 32. If he writes the list out, fine. If he says 35, he has counted numbers instead of jumps.
• “Check your method on the 3rd number.” Does it give back 11? This is the habit, so see whether he does it without being told.
• Draw three squares in a row and ask how many sticks 5 squares would need. The answer is 16. Watch whether he looks for what is new each time.
• Write 2, 5, 4, 7, 6, 9 and ask what comes next. The next number is 8. If he says 8, he has seen the two strands. If he is stuck, he is reading the numbers as one line, and reading every other number is the lesson.

If any of these goes wrong, the usual cause is not the arithmetic. It is the same quiet thing as in when a child guesses instead of working it out: he has a rule that feels right and has never been asked to prove it.

What does not help

More lists to continue. They practise the easy half, which he already has, and they let him believe the topic is secure.

And teaching the formula early. A child who writes “3n + 2” without knowing where the 2 comes from will use it confidently on the next pattern too, including the ones where it does not apply.

The short version

• Continuing a pattern and finding the 50th term are different skills. Most children have only the first.
• To reach the 50th term there are 49 jumps, not 50. Count the gaps, not the numbers.
• Check any rule on a term you can already see before using it on the far one.
• In a shape pattern, ask what is new each time and what was always there.
• Write the gaps underneath first. They tell you whether to add, multiply, or split the pattern in two.
• Build the rule from the jumps before ever giving him the shortcut.

Word problems that hide a pattern call for the same habit of slowing down to ask what is being repeated; see when the sums are fine but the story is not.