Patterns and Rules

Number Patterns

A pattern is a sequence that follows a rule.

Example: 3, 6, 9, 12, 15, ...

• Rule: Add 3 each time
• Next numbers: 18, 21, 24

Writing Rules

| Pattern | Rule | |---------|------| | 2, 4, 6, 8, ... | Add 2 | | 5, 10, 20, 40, ... | Multiply by 2 | | 100, 90, 80, 70, ... | Subtract 10 | | 1, 4, 9, 16, 25, ... | Square numbers ($1^2, 2^2, 3^2, ...$) |

Input-Output Tables

A rule connects input numbers to output numbers:

| Input | Output | |-------|--------| | 1 | 5 | | 2 | 8 | | 3 | 11 | | 4 | ? |

Rule: Multiply by 3, then add 2. So input 4 → $4 \times 3 + 2 = 14$

Features of Patterns

• Patterns can be increasing (going up) or decreasing (going down)
• The same rule always gives the same result
• Patterns help us predict what comes next

Even and Odd Patterns

• Even + Even = Even: $4 + 6 = 10$
• Odd + Odd = Even: $3 + 5 = 8$
• Even + Odd = Odd: $4 + 3 = 7$

Challenge: What rule makes this pattern? 1, 1, 2, 3, 5, 8, 13, ... (Answer: Add the two previous numbers — this is the Fibonacci sequence!)