Number Patterns - Finding Rules and nth Terms
A number pattern is a list of numbers arranged according to a rule. Recognising the rule lets you extend the pattern, find missing terms, and describe the relationship between numbers in precise mathematical language.
The study of number patterns is one of the oldest branches of mathematics: the ancient Greek philosopher Pythagoras and his followers, around the 6th century BCE, were fascinated by "figurate numbers" – numbers that can be arranged into geometric shapes, such as the triangular and square numbers still taught today. They believed these patterns revealed a deep hidden order in the universe, literally worshipping numbers as sacred. That same instinct to spot an underlying rule now drives fields far beyond ancient philosophy: computer scientists analysing algorithm performance describe how running time grows using exactly the same nth-term thinking taught on this page, and financial analysts scan stock data for repeating patterns that might predict future prices.
What Makes a Number Pattern?
Every number pattern has an underlying rule that connects each term to the next (or to its position in the list). Finding that rule is the key skill. Once you have it, you can generate any term in the sequence without listing all the ones before it.
Common Types of Number Pattern
| Type | Rule | Example |
|---|---|---|
| Adding a fixed amount | Add the same number each time | 3, 7, 11, 15, 19 … (add 4) |
| Subtracting a fixed amount | Subtract the same number each time | 50, 44, 38, 32 … (subtract 6) |
| Multiplying by a fixed amount | Multiply by the same number each time | 2, 6, 18, 54 … (multiply by 3) |
| Dividing by a fixed amount | Divide by the same number each time | 256, 64, 16, 4 … (divide by 4) |
| Square numbers | n² | 1, 4, 9, 16, 25 … |
| Cube numbers | n³ | 1, 8, 27, 64, 125 … |
| Triangular numbers | Add 1, then 2, then 3 … | 1, 3, 6, 10, 15 … |
| Powers of 2 | Double each time | 1, 2, 4, 8, 16, 32 … |
Finding the nth Term Rule
The nth term (or general term) is a formula that gives the value of any term directly from its position number n.
- For add/subtract patterns: nth term = first term + (n − 1) × common difference.
- For multiply/divide patterns: nth term = first term × (common ratio)^(n−1).
- For square numbers: nth term = n².
Worked Examples
Differences: 9−5=4, 13−9=4, 17−13=4. Rule: add 4 each time.
Next terms: 17+4 = 21, 21+4 = 25.
Check ratio: 12÷3 = 4, 192÷48 = 4 (checking 48). 12 × 4 = 48. Rule: multiply by 4.
First term = 6. Common difference = 5. nth term = 6 + (n−1) × 5 = 6 + 5n − 5 = 5n + 1.
Check: n=1 → 6 ✓; n=3 → 16 ✓.
5n + 1 = 101 → 5n = 100 → n = 20. The 20th term equals 101.
Key Takeaways
- Every number pattern follows a rule – find differences or ratios to identify it.
- The nth term formula lets you find any term without listing all previous ones.
- For linear (add/subtract) patterns: nth term = a + (n−1)d, where a = first term, d = common difference.
- Always verify your rule by checking it against known terms.