Number Patterns & Sequences

Arithmetic Sequences

Each term is found by adding (or subtracting) a fixed number called the common difference (d).

Example: 3, 7, 11, 15, … → d = 4

Formula for the nth term: T(n) = a + (n−1)d
where a = first term, d = common difference

Example: Find the 10th term of 3, 7, 11, 15, …
T(10) = 3 + (10−1) × 4 = 3 + 36 = 39

Geometric Sequences

Each term is found by multiplying by a fixed number called the common ratio (r).

Example: 2, 6, 18, 54, … → r = 3

Finding the Rule

Look at the difference between consecutive terms:

• Constant difference → arithmetic sequence
• Constant ratio → geometric sequence
• Differences form their own pattern → quadratic or other sequence

Example: 1, 4, 9, 16, 25, … → these are perfect squares (n²)

Shape/Matchstick Patterns

Many pattern questions use visual sequences. Count carefully, find the rule, then apply it.

Example: A pattern uses 4, 7, 10, 13 matchsticks for 1, 2, 3, 4 squares.
Rule: T(n) = 3n + 1. For 10 squares: T(10) = 31 matchsticks.

Key Tips

• Always find the pattern rule, not just the next term
• Draw a table with n (term number) and T(n) (term value) to spot patterns
• Check your rule by substituting n = 1, 2, 3 and seeing if it matches