# Number Patterns and Rules

> Find the next term and the rule of a pattern, use square, cube and doubling patterns, and read Roman numerals.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/number-patterns/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Birthday money that doubles
Grandma has a strange birthday tradition. On Lily's 1st birthday she gave her 1 dollar. On her 2nd birthday, 2 dollars. On her 3rd, 4 dollars. Each year the gift doubles.

"When will I get more than 2000 dollars?" Lily wonders. She lists them: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048. The 12th number is 2048, so on her 12th birthday she gets over 2000 dollars!

Lily spotted the **pattern** (double each time) and used it to look far ahead. That is what number patterns are for.

**In short:** To continue a number pattern, find how each term changes: by adding, subtracting, multiplying or dividing the same amount. If the jumps themselves change, look for a pattern in the jumps.

## Before you start

- Times tables and adding.
- Powers: $2^3 = 2 \times 2 \times 2 = 8$ and $10^2 = 100$.
- Substituting a number into a rule like $y = 2x + 1$ (see Substitution in the Algebra unit).

## Spotting the pattern

Look at how you get from one term to the next.

| Pattern | What happens | Next |
|---|---|---|
| 3, 7, 11, 15, ... | add 4 | 19 |
| 50, 44, 38, 32, ... | subtract 6 | 26 |
| 2, 6, 18, 54, ... | multiply by 3 | 162 |
| 1, 4, 9, 16, 25, ... | square numbers $1^2, 2^2, 3^2$ | 36 |
| 1, 8, 27, 64, ... | cube numbers $1^3, 2^3, 3^3$ | 125 |
| 1, 1, 2, 3, 5, 8, ... | add the two before | 13 |

### Exam tip: Write the gaps underneath
If the jumps are not all the same, write the jumps under the pattern. Sometimes the jumps themselves make a pattern.

2, 3, 5, 8, 12, ... has jumps 1, 2, 3, 4, so the next jump is 5 and the next term is 17.

## The rule for any term

If a pattern goes up by the **same amount** each time, the rule is

$$\text{term} = \text{jump} \times n + \text{something}$$

### Worked example: Find the rule
5, 8, 11, 14, ... The jump is 3, so try $3n$. When $n = 1$, $3n = 3$, but the term is 5. We need 2 more. Rule: $3n + 2$.

The 20th term is $3 \times 20 + 2 = 62$.

## Rules with squares

A rule can use a square. Just substitute carefully.

### Worked example: Filling a table
Rule: $y = 2x^2 + 1$

| $x$ | 3 | 4 | 5 | 10 |
|---|---|---|---|---|
| $y$ | 19 | 33 | 51 | 201 |

For $x = 3$: $2 \times 9 + 1 = 19$.

### Worked example: Brackets first
Rule: $t = (s - 2)^2 + 4$. When $s = 7$: $(7 - 2)^2 + 4 = 25 + 4 = 29$.

Rule: $r = q^2 - q$. When $q = 10$: $100 - 10 = 90$.

## Roman numerals

Some puzzles use Roman numerals.

| I | V | X | L | C | D | M |
|---|---|---|---|---|---|---|
| 1 | 5 | 10 | 50 | 100 | 500 | 1000 |

### Key idea: How to read them
Add the values, biggest first: $\text{LX} = 50 + 10 = 60$ and $\text{LV} = 55$.

A **smaller** numeral **before** a bigger one is **taken away**: $\text{IV} = 4$, $\text{IX} = 9$, $\text{XL} = 40$.

So $\text{XLV} = 40 + 5 = 45$.

## Cross-number puzzles

In a cross-number puzzle, each row and each column follows the operation sign shown, such as $+$ or $\times$. Work out the boxes you know first, then use them to find the empty ones.

### Worked example: A small cross-number
The puzzle uses $+$. The top row is 59 and 98. Down the first column: 59 and $\frac{1}{4} \times 10^2 = 25$. So the bottom-left total is $59 + 25 = 84$. Across the top: $59 + 98 = 157$.

### Challenge: Selective level: doubling and growing patterns
Doubling grows very fast: $2^{10} = 1024$. That is why Lily's 12th birthday gift, $2^{11} = 2048$ dollars, is over 2000.

Matchstick squares in a row: 1 square uses 4 sticks, 2 squares use 7, 3 squares use 10. The jump is 3, so the rule is $3n + 1$. For 50 squares you need $3 \times 50 + 1 = 151$ sticks.

Triangular numbers 1, 3, 6, 10, 15, ... are $1$, $1 + 2$, $1 + 2 + 3$, and so on. The 10th one is $1 + 2 + \dots + 10 = 55$.

**Mind map**

```text
Number patterns
  Find the change
    Add or subtract
    Multiply or divide
    Look at the jumps
  Special patterns
    Squares 1, 4, 9, 16
    Cubes 1, 8, 27
    Doubling 1, 2, 4, 8
  nth term
    jump x n + something
    $3n + 2$
  Roman numerals
    I V X L C D M
    Small before big: take away
```

## Quick summary

- Find how each term changes: add, subtract, multiply or divide.
- If the jumps change, write them down and look for a pattern in the jumps.
- A pattern that grows by the same jump has a rule like $3n + 2$.
- Know the square numbers, cube numbers and doubling numbers.
- Roman numerals: I 1, V 5, X 10, L 50, C 100. A smaller one before a bigger one is taken away.

## Quiz

Try these questions from the chapter quiz.

1. What is the next number? 3, 7, 11, 15, ...
   - A. 18
   - B. 19
   - C. 20
   - D. 21
   - Answer: B. 19. The pattern adds 4 each time: $15 + 4 = 19$.

2. What is the next square number after 49?
   - A. 50
   - B. 56
   - C. 64
   - D. 81
   - Answer: C. 64. $7^2 = 49$, so the next is $8^2 = 64$.

3. What is LX in our numbers?
   - A. 60
   - B. 40
   - C. 510
   - D. 15
   - Answer: A. 60. L is 50 and X is 10. Bigger first, so add: 60.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/number-patterns/#quiz