Recognising patterns and expressing them as rules is a foundational mathematical skill. In Foundation Mathematics, this means identifying how a sequence of numbers changes, writing a rule to describe it, and using the rule to find terms or solve practical problems.
KEY TAKEAWAY: A rule for a number sequence lets you find any term without listing every value. Finding the rule is the key step.
Step 1: Calculate the differences between consecutive terms.
$\$5, 9, 13, 17, 21, \ldots$$
$$\text{Differences: } +4, +4, +4, +4 \quad \Rightarrow \text{Arithmetic (linear)}$$
Step 2: If differences are not constant, check ratios.
$\$3, 6, 12, 24, 48, \ldots$$
$$\text{Ratios: } \times 2, \times 2, \times 2 \quad \Rightarrow \text{Geometric (exponential)}$$
Step 3: If neither, look for second-level differences.
$\$1, 4, 9, 16, 25, \ldots$$
$$\text{First differences: } 3, 5, 7, 9 \quad \text{Second differences: } 2, 2, 2 \quad \Rightarrow \text{Quadratic (rule involves }n^2\text{)}$$
For an arithmetic sequence with first term $t_1$ and common difference $d$:
$$\boxed{t_n = t_1 + (n-1) \times d}$$
Worked Example 1 — Find the rule:
Sequence: \$7, 11, 15, 19, \ldots$
$$d = 11 - 7 = 4, \quad t_1 = 7$$
$$t_n = 7 + (n-1) \times 4 = 7 + 4n - 4 = 4n + 3$$
Check: $n = 1: 4(1)+3 = 7$ ✓, $n = 3: 4(3)+3 = 15$ ✓
Worked Example 2 — Use the rule:
Using the above sequence, find the 20th term.
$$t_{20} = 4(20) + 3 = 80 + 3 = 83$$
Worked Example — Does a given value appear in the sequence?
Is $95$ in the sequence \$7, 11, 15, 19, \ldots$?
$$4n + 3 = 95$$
$$4n = 92$$
$$n = 23$$
Since $n = 23$ is a whole number, yes — $95$ is the 23rd term.
If $n$ were not a whole number (e.g. $n = 7.5$), the value would NOT be in the sequence.
A rule can be represented as a table or graph.
Rule: Cost of printing = $\$0.15$ per page + $\$2.00$ setup fee
$$\text{Cost} = 0.15n + 2.00$$
| Pages ($n$) | Cost ($\$$) |
|---|---|
| 0 | 2.00 |
| 10 | 3.50 |
| 20 | 5.00 |
| 50 | 9.50 |
Graphing this produces a straight line — the $y$-intercept is $2.00$ (setup fee) and the gradient is $0.15$ (cost per page).
EXAM TIP: If a relationship produces a straight-line graph, the rule is linear. The gradient is the rate of change; the $y$-intercept is the starting value.
Patterns can also appear in arrangements of shapes (matchstick problems).
Example: A row of squares made from matchsticks.
- 1 square → 4 matches
- 2 squares → 7 matches
- 3 squares → 10 matches
$$\text{Differences: } +3, +3 \quad \Rightarrow \text{Arithmetic, } d = 3$$
$$t_n = 4 + (n-1) \times 3 = 3n + 1$$
Check: $n = 4: 3(4) + 1 = 13$ matches ✓
APPLICATION: Pattern rules appear throughout Foundation Mathematics — in financial calculations, tiling costs, planting arrangements, and time-based growth problems. Recognising a sequence as arithmetic is the first step to writing a formula that saves significant calculation time.
