Number Sequences
Year 7 📊 Algebra Find patterns, nth terms and describe arithmetic sequences.
🔢 Term-to-Term Rules
A sequence is an ordered list of numbers. A term-to-term rule tells you how to get from one term to the next.
Example: 3, 7, 11, 15, … → add 4 each time. This is an arithmetic sequence with common difference 4.
Example: 2, 6, 18, 54, … → multiply by 3 each time. This is a geometric sequence with common ratio 3.
📐 The nth Term Formula
The nth term formula lets you find any term directly without listing all previous terms.
⚡ Arithmetic nth Term
$$T_n = a + (n-1)d$$where $a$ = first term, $d$ = common differenceExample: Sequence 5, 8, 11, 14, … has $a=5$, $d=3$. nth term $= 5 + (n-1) \times 3 = 3n + 2$. Check: when $n=1$: $3(1)+2=5$ ✓
🔍 Special Sequences
Some sequences have well-known patterns worth recognising.
| Name | Sequence | nth term |
|---|---|---|
| Square numbers | 1, 4, 9, 16, 25, … | $n^2$ |
| Cube numbers | 1, 8, 27, 64, … | $n^3$ |
| Triangle numbers | 1, 3, 6, 10, 15, … | $\frac{n(n+1)}{2}$ |
| Fibonacci | 1, 1, 2, 3, 5, 8, … | Each term = sum of previous two |
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Interactive Demonstration — Number Sequences
🔢 Sequence Explorer