What you'll learn
A number pattern, or sequence, is a list of numbers arranged by a rule, and the whole of this topic is about finding that rule and then using it. In the pattern 5, 9, 13, 17, each number is 4 more than the one before, so the rule is "add 4" and the next number must be 21. This guide covers finding the term-to-term rule, continuing a pattern forwards and backwards, filling in a missing number in the middle, working out a far-off term without writing out the whole list, and recognising the special patterns the SEA paper reuses year after year — odd and even numbers, square numbers, triangular numbers and doubling patterns. Pattern questions appear in the Number strand, which is 19 of the 40 items on the SEA Mathematics paper, and they are usually among the quickest marks on it once the method is secure.
Key terms and definitions
Sequence (pattern) — a list of numbers that follows a rule, such as 2, 4, 6, 8, 10.
Term — one number in the sequence. In 2, 4, 6, 8 the third term is 6.
Position — where a term sits in the list: first, second, third and so on.
Term-to-term rule — what you do to one term to get the next, such as "add 4" or "multiply by 2".
Position rule — a rule that gets you straight from the position number to the term, such as "multiply the position by 4, then add 1".
Difference — the gap between one term and the next.
Ascending — going up. Descending — going down.
Consecutive — following on one after another with nothing in between.
Triangular numbers — 1, 3, 6, 10, 15, 21, 28 — the counts of dots that make growing triangles.
Core concepts
Always find the differences first
The single most useful habit in this topic is to write the gaps underneath the numbers before doing anything else.
For 7, 12, 17, 22, 27 the gaps are 5, 5, 5, 5. The rule is "add 5", and the next term is 32.
For 100, 91, 82, 73 the gaps are −9, −9, −9. The rule is "subtract 9", and the next term is 64.
When every gap is the same, the pattern is a constant-difference pattern, and these make up most of what the SEA paper asks. Writing the gaps down converts a puzzle into a fact.
When the differences are not equal
If the gaps are not the same, check three things in order.
Is it a multiplying pattern? In 3, 6, 12, 24, 48 the gaps are 3, 6, 12, 24 — not equal — but each term is double the one before. The rule is "multiply by 2". Multiplying patterns grow very fast, which is the clue to look for.
Are the differences themselves a pattern? In 1, 3, 6, 10, 15 the gaps are 2, 3, 4, 5. The gaps go up by one each time, so the next gap is 6 and the next term is 21. These are the triangular numbers.
Is the rule alternating? In 1, 2, 4, 5, 7, 8 the gaps are 1, 2, 1, 2. The rule is "add 1, then add 2, and repeat", so the next term is 10.
Working backwards
A pattern can be extended in the other direction by undoing the rule. If the rule is "add 6" and the sequence starts 19, 25, 31, the term before 19 is 19 − 6 = 13, and the one before that is 7.
This matters because some SEA items give you the middle of a sequence and ask for the first term. The instinct is to guess; the method is to reverse the rule.
Filling a gap in the middle
If a term is missing, use the terms on both sides of it as a check.
In 8, 15, __, 29, 36 the gaps that you can see are 7 (from 8 to 15) and 7 (from 29 to 36). So the rule is "add 7", and the missing term is 15 + 7 = 22. Checking the other way: 29 − 7 = 22. ✔ Both checks agree, so the answer is safe.
Always check a filled gap from both sides. It takes a moment and it catches every careless slip.
Getting to a far-off term without listing everything
If a question asks for the 20th term, writing out twenty numbers is slow and one slip ruins it. Use the position rule instead.
Take 5, 9, 13, 17. The difference is 4, so the position rule starts with "multiply the position by 4". Test it on the first term: 1 × 4 = 4, but the first term is 5, so you need one more. The rule is "multiply the position by 4, then add 1".
Check it on another term: the fourth term should be 4 × 4 + 1 = 17. ✔
Now the 20th term is 20 × 4 + 1 = 81, found in one line.
The pattern is always the same. The difference tells you what to multiply by; then adjust with a plus or minus so that the first term comes out right.
The special patterns worth memorising
| Pattern | First terms | Rule |
|---|---|---|
| Even numbers | 2, 4, 6, 8, 10 | add 2 |
| Odd numbers | 1, 3, 5, 7, 9 | add 2 |
| Square numbers | 1, 4, 9, 16, 25, 36, 49 | position × position |
| Triangular numbers | 1, 3, 6, 10, 15, 21, 28 | add 2, then 3, then 4 … |
| Doubling | 1, 2, 4, 8, 16, 32, 64 | multiply by 2 |
| Multiples of 5 | 5, 10, 15, 20 | add 5 |
If a sequence in an exam looks strange, compare it with this table before working hard. Very often it is one of these with a small change — 2, 5, 10, 17 is simply the square numbers with one added to each.
Patterns made from shapes
Some items build a pattern out of sticks, tiles or dots and ask how many are needed for a later shape. The method is exactly the same: turn the shapes into numbers, then find the rule.
If Kern builds a row of squares with matchsticks, one square takes 4 sticks, two squares take 7, and three squares take 10.
| Squares | Sticks |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
| 4 | 13 |
The difference is 3, because each extra square only needs three new sticks — it borrows one side from the square before it. So the position rule is "multiply the number of squares by 3, then add 1". Checking: 3 × 3 + 1 = 10. ✔
Worked examples
Example 1: Continuing a pattern and finding a far term
Look at the pattern 5, 9, 13, 17, … Write down the next two terms and work out the 10th term.
First write the gaps: 9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4. The rule is "add 4".
The next two terms are 17 + 4 = 21 and 21 + 4 = 25.
For the 10th term, do not list them all. The difference is 4, so try "position × 4". For position 1 that gives 4, but the first term is 5, so add 1. The position rule is "position × 4 + 1".
Test it on the third term: 3 × 4 + 1 = 13. ✔
So the 10th term is 10 × 4 + 1 = 41.
Example 2: A matchstick pattern
Anisa makes a row of squares out of matchsticks. One square uses 4 sticks, two squares use 7 sticks and three squares use 10 sticks. How many sticks does she need for 8 squares?
Write the numbers out: 4, 7, 10. The gaps are 3 and 3, so each extra square costs 3 sticks.
The position rule is "number of squares × 3, then add 1". Check it on two squares: 2 × 3 + 1 = 7. ✔
For 8 squares: 8 × 3 + 1 = 24 + 1 = 25 sticks.
If you prefer to continue the list, it runs 4, 7, 10, 13, 16, 19, 22, 25 — the eighth number is 25, which agrees.
Example 3: A missing term where the gaps change
Find the missing number: 1, 3, 6, 10, __, 21, 28.
The gaps are 2, 3, 4, then something, then something. They are going up by one each time.
So the gap from 10 to the missing term must be 5, giving 10 + 5 = 15.
Check from the other side: the next gap should be 6, and 15 + 6 = 21. ✔ Then 21 + 7 = 28. ✔
These are the triangular numbers, and they are worth recognising on sight because they appear regularly.
Common mistakes and how to avoid them
Guessing the rule from the first two terms only. Deciding that 2, 4, … is doubling when it is actually adding 2. Fix: check the rule against at least three gaps before using it.
Not writing the gaps down. Trying to spot the rule by staring at the numbers. Fix: write the differences underneath in pencil, every time.
Getting the direction wrong in a descending pattern. Adding when the sequence is going down. Fix: if the numbers are getting smaller, the rule must take something away.
Confusing the term with its position. Answering "10" when asked for the 10th term. Fix: circle the position number in the question, and say "the tenth one is …" out loud.
Forgetting the adjustment in a position rule. Saying the rule for 5, 9, 13 is just "×4" and getting 40 for the 10th term. Fix: always test the rule on the first term before you trust it.
Filling a middle gap from one side only. Fix: check from the left and from the right; the two must agree.
Assuming the rule is always "add". Fix: if the numbers grow fast, test multiplying instead.
How parents can help at home
Patterns are the friendliest topic on the paper for a parent to help with, because you can check the answer even if you have forgotten every method.
Ask "what comes next, and how do you know?" The second half of that question is the one that matters. A child who can say "because it goes up by 4 each time" has done the actual work.
Make patterns out of real things. Lay out spoons, seeds or bottle caps in rows of 3, 5, 7 and ask how many for the next row. Turning objects into numbers is exactly what the shape-pattern items require.
Count in steps in the car. Count on in 6s, then backwards in 6s from 60. Counting backwards is the weaker skill in most children and it is what descending patterns need.
When your child is stuck, say "write the differences underneath". That prompt alone solves most pattern questions, and you do not have to know the answer to give it.
Practise the special lists. Squares to 144 and triangular numbers to 55 are quick to learn and instantly recognisable in an exam.
Exam technique for number patterns
The SEA Mathematics paper is weighted 100 marks, against 60 for ELA and 40 for ELA Writing, and it has 40 items worth 75 marks. With Number carrying 19 items, patterns turn up most years, usually as one or two multiple-choice items.
Write the differences on the paper even in a multiple-choice item. It is faster than thinking about it, and the working space is provided.
Check any rule against at least two terms before you use it to find a later one. A rule that fits only the first pair will send you to a wrong option that has been placed there on purpose.
For a far-off term, use the position rule rather than listing. Listing twenty terms invites one small slip that ruins the whole answer.
When a question shows shapes, make a small table of numbers first. The pattern is almost never visible in the pictures but it is obvious in the table.
Read the question once more before answering. Some items ask for the next term, some for the rule, and some for the term at a given position — three different answers from the same sequence.
Quick revision summary
- Write the differences between terms before anything else.
- Equal differences mean an "add" or "subtract" rule.
- Fast growth suggests a multiplying rule such as doubling.
- If the differences themselves form a pattern, look at triangular numbers.
- Reverse the rule to extend a pattern backwards.
- Check a missing middle term from both sides — the two checks must agree.
- For a far-off term, use a position rule: difference × position, then adjust.
- Test any position rule on the first term before trusting it.
- Know by heart: squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
- Know by heart: triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.
- For shape patterns, turn the shapes into a table of numbers first.