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 \(d\):
Worked Example 1 — Find the rule:
Sequence: \(7, 11, 15, 19, \ldots\)
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 \(7, 11, 15, 19, \ldots\)?
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
| 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
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.
