Number Patterns and Rules
Discover the rules hiding inside number sequences. Once you know the rule, you can predict any term, even the millionth one, instantly.
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Before you read on, quickly: Look at this pattern:
2, 5, 8, 11, 14, ?, ?
What comes next? What is the rule? Can you find it before reading on?
Patterns are everywhere in mathematics. A sequence is a list of numbers that follows a rule. Once we know the rule, we can predict any term in the sequence.
There are two main types of rules:
- Term-to-term rule: Tells us how to get from one term to the next. For example, “add 3.”
- Position-to-term rule: Links the position number directly to the term value. For example, “multiply position by 2 and add 1.”
These rules are the building blocks of understanding linear relationships.
Know
- Number patterns follow rules that describe how terms are related
- A term-to-term rule describes how to get from one term to the next
- A position-to-term rule links the position number directly to the term value
Understand
- A single rule can generate an entire infinite sequence
- How tables of values organise pattern information systematically
- Position-to-term rules are more powerful than term-to-term rules
Can Do
- Continue patterns given a term-to-term rule
- Find the term-to-term rule from a given sequence
- Create tables of values from position-to-term rules
- Plot pattern points on the Cartesian plane
Wrong: “The term-to-term rule is $2n + 1$.” That formula uses a position number $n$, it is a position-to-term rule, not term-to-term!
Right: Term-to-term describes the change between terms (e.g., “add 2”). Position-to-term gives a formula for any term directly (e.g., $y = 2n + 1$).
Wrong: Checking only one difference and assuming that is the rule: $3, 6, 12, 24$, seeing $6 - 3 = 3$ and saying “add 3.” But $12 - 6 = 6$!
Right: Always check at least two or three differences before declaring the rule. The real rule for $3, 6, 12, 24$ is “multiply by 2.”
A term-to-term rule tells you how to get from one term to the next. It describes the change between consecutive terms.
- Calculate the difference between consecutive terms: $7 - 3 = 4$, $\ 11 - 7 = 4$, $\ 15 - 11 = 4$.
- Check that the difference is the same each time. Yes, each difference is $4$.
- Write the rule: “Add 4”. The next two terms would be $15 + 4 = 19$ and $19 + 4 = 23$.
Using a position-to-term rule is even more powerful because it lets you find any term without listing all the ones before it.
- Substitute $n = 1$: $y = 2(1) + 1 = 3$. So the 1st term is $3$.
- Substitute $n = 2, 3, 4, 5$:
$n=2$: $y = 2(2)+1 = 5$ $n=3$: $y=7$ $n=4$: $y=9$ $n=5$: $y=11$. - The sequence is $3, 5, 7, 9, 11, \ldots$ The term-to-term rule is “add 2”, and the position-to-term rule is $y = 2n + 1$.
| Position ($n$) | Calculation ($2n+1$) | Term |
|---|---|---|
| 1 | $2 \times 1 + 1$ | 3 |
| 2 | $2 \times 2 + 1$ | 5 |
| 3 | $2 \times 3 + 1$ | 7 |
| 4 | $2 \times 4 + 1$ | 9 |
| 5 | $2 \times 5 + 1$ | 11 |
A table of values organises pairs of related numbers systematically. It is the bridge between patterns, rules, and graphs.
- Set up the table with columns for $x$ and $y = 3x + 2$.
- Substitute each $x$ value: $x=0 \to 2$; $x=1 \to 5$; $x=2 \to 8$; $x=3 \to 11$; $x=4 \to 14$.
- Notice: the $y$ values increase by 3 each time ($5-2=3$, $8-5=3$, etc.). This equals the coefficient of $x$ in the rule!
| $x$ | $y = 3x + 2$ |
|---|---|
| 0 | 2 |
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
| 4 | 14 |
We can visualise the pattern by plotting each $(x, y)$ pair on the Cartesian plane. The position number becomes the $x$-coordinate; the term value becomes the $y$-coordinate.
Plotting $(x, y)$ pairs from $y = 2x + 1$, the points form a straight line!
Common Pitfalls
- Confusing term-to-term with position-to-term: Term-to-term says “how to move” (e.g., “add 2”). Position-to-term says “where you are” (e.g., $y = 2n + 1$). Ask yourself: does my rule use a position number?
- Thinking the starting value comes from the rule: Both $3, 6, 9, 12$ and $7, 10, 13, 16$ have the rule “add 3” but different starting values. You need both to define a sequence.
- Checking only one difference: Always check at least two or three differences. For $3, 6, 12, 24$: $6-3=3$ but $12-6=6$, the rule is “multiply by 2”, not “add 3”.
Copy Into Books
Finding a Term-to-Term Rule
- Subtract consecutive terms to find the differences
- Check all differences are the same
- State the rule (e.g., “add 4” or “subtract 5”)
Position-to-Term Rules
- A formula using $n$ gives any term directly
- For $y = mx + c$: $y$ increases by $m$ each time $x$ increases by 1
- The starting value (when $x = 0$) is $c$
Plotting Pattern Points
- Use position as $x$-coordinate, term as $y$-coordinate
- Points from $y = mx + c$ always form a straight line
- This is called a linear relationship
How are you completing this lesson?
Brain Trainer · 4 problems
Four drill problems to sharpen your pattern skills. Work each, then reveal the answer.
-
1 Find the next term: $7, 12, 17, 22, \ldots$
The common difference is $12 - 7 = 5$. Add 5 to 22.27 -
2 Find the 4th term using the rule $y = 3n + 2$.
Substitute $n = 4$: $y = 3(4) + 2 = 12 + 2$.14 -
3 A function machine doubles the input then subtracts 1. Input is 5. Output?
Rule: $y = 2x - 1$. Substitute $x = 5$: $y = 2(5) - 1 = 10 - 1$.9 -
4 Position 3, term 8. What are the coordinates when plotting this pattern point?
Position is the $x$-coordinate; term is the $y$-coordinate.$(3, 8)$
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Find the next 3 terms and state the term-to-term rule for the sequence: $1, 4, 7, 10, \ldots$
Q7. A pattern starts at 2 and each term is multiplied by 3. Write the first 5 terms of this sequence. What type of rule is this?
Q8. Complete the table of values for the rule $y = 2x + 3$ with $x = 0, 1, 2, 3, 4$. Describe what you notice about the $y$ values.
Quick Check
1. B Add 4. All differences ($9-5$, $13-9$, $17-13$) equal 4.
2. D14. Rule $y = 3n+2$; for $n = 4$: $3(4)+2 = 14$.
3. C$1, 5, 9, 13$. Check: $5-1=4$, $9-5=4$, $13-9=4$. Consistent.
4. C9. Rule $y = 2x-1$; for $x=5$: $2(5)-1=9$.
5. B$(3, 8)$. Position is $x$, term is $y$: coordinates $(3, 8)$.
Show Your Working Model Answers
Q6 (3 marks): Differences: $4-1=3$, $7-4=3$, $10-7=3$ [1]. Rule: “add 3” [1]. Next 3 terms: $13, 16, 19$ [1].
Q7 (3 marks): Apply “multiply by 3” repeatedly: $2, 6, 18, 54, 162$ [2]. This is a term-to-term rule (also called geometric) because it describes how to move from one term to the next [1].
Q8 (3 marks): Table: $x=0\to3$; $x=1\to5$; $x=2\to7$; $x=3\to9$; $x=4\to11$ [2]. What I notice: the $y$ values increase by 2 each time (matching the coefficient of $x$), and the starting value when $x=0$ is 3 (matching the constant) [1].
Finding the nth Term
Challenge 1: The sequence $5, 8, 11, 14, \ldots$ has the position-to-term rule $y = 3n + 2$.
(a) Find the 10th term. (b) Find the 20th term. (c) Which position gives the term 50? (Hint: solve $3n + 2 = 50$.)
Challenge 2: Find the position-to-term rule for the sequence $4, 7, 10, 13, 16, \ldots$
(Hint: the common difference is 3, so the rule starts $y = 3n + \ldots$ Find the constant by substituting $n = 1$.)
Challenge 3: The $n$th term of a sequence is $4n - 1$.
(a) Find the first 4 terms. (b) Is 99 a term in this sequence? (Hint: solve $4n - 1 = 99$.)
Reveal solutions
C1: (a) $3(10)+2 = 32$ (b) $3(20)+2 = 62$ (c) $3n+2=50 \Rightarrow 3n=48 \Rightarrow n=16$. The 16th term is 50.
C2: Common difference $= 3$, so $a = 3$. Try $y = 3n + b$. For $n=1$, $y=4$: $4 = 3(1) + b \Rightarrow b = 1$. Rule: $y = 3n + 1$. Check: $3(2)+1=7$ ✓.
C3: (a) $4(1)-1=3$; $4(2)-1=7$; $4(3)-1=11$; $4(4)-1=15$. First 4 terms: $3, 7, 11, 15$. (b) $4n-1=99 \Rightarrow 4n=100 \Rightarrow n=25$. Since $n=25$ is a whole number, 99 IS in the sequence (the 25th term).
Term-to-term rule
How to move from one term to the next (e.g., “add 3”)
Position-to-term rule
Formula using $n$ to find any term directly
Function machine
Input → rule → output. Rule is the position-to-term formula.
Table of values
Systematic list of input-output pairs from the rule
Check all differences
Always verify at least 2–3 gaps before declaring the rule
Linear pattern
Points from $y = mx + c$ always form a straight line
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