Sequences & Patterns

Types of Sequences

A sequence is an ordered list of numbers that follow a pattern. Each number in the list is called a term.

Arithmetic Sequences

An arithmetic sequence has a constant common difference (d) between consecutive terms — you add (or subtract) the same amount each time.

• 3, 7, 11, 15, … — common difference: +4
• 20, 14, 8, 2, … — common difference: -6

To find d: subtract any term from the term that follows it.

Geometric Sequences

A geometric sequence has a constant common ratio (r) between consecutive terms — you multiply by the same number each time.

• 3, 6, 12, 24, … — common ratio: ×2
• 100, 50, 25, 12.5, … — common ratio: ×0.5 (halving each time)

To find r: divide any term by the term before it.

Other Sequences

• Square numbers: 1, 4, 9, 16, 25, … (nth term = n²)
• Cube numbers: 1, 8, 27, 64, 125, … (nth term = n³)
• Triangular numbers: 1, 3, 6, 10, 15, … (add one more each time)
• Fibonacci: 1, 1, 2, 3, 5, 8, 13, … (each term is the sum of the previous two)

Term-to-Term Rule

A term-to-term rule describes how to get from one term to the next. For example: "start at 3, add 4 each time."

This is useful for continuing a sequence but requires you to list all earlier terms to reach a specific one.

Position-to-Term Rule (nth Term)

A position-to-term rule (nth term formula) gives the value of any term directly from its position number n.

This is more powerful — you can find the 100th term without listing the first 99.

Finding the nth Term of an Arithmetic Sequence

Formula: aₙ = a + (n - 1)d

Where:

• a = first term
• d = common difference
• n = position number

A simplified version: if the common difference is d and the sequence starts at position 1, the nth term has the form dn + c, where c is a constant.

Worked Example: Find the nth term of 5, 8, 11, 14, …

Step 1: Find d = 8 - 5 = 3, so the formula starts with 3n

Step 2: When n = 1, term = 5. But 3(1) = 3. We need to add 2 to get 5.

Step 3: nth term = 3n + 2

Check: n = 1 → 3(1) + 2 = 5 ✓ | n = 2 → 3(2) + 2 = 8 ✓ | n = 4 → 3(4) + 2 = 14 ✓

Finding a Specific Term

Substitute the position number into the nth term formula.

Example: 10th term of the sequence with nth term 4n - 1:

4(10) - 1 = 40 - 1 = 39

Finding Which Term Has a Given Value

Set the nth term equal to the target value and solve for n.

Example: Which term of 3n + 1 equals 31?

3n + 1 = 31 → 3n = 30 → n = 10

So 31 is the 10th term.

Recognising Sequence Types

Pattern	Type
Add or subtract same amount	Arithmetic
Multiply or divide by same amount	Geometric
Differences increase by 1 each time	Triangular numbers
Differences are 2, 4, 6, 8, …	Square numbers
Each term is sum of previous two	Fibonacci

Common Mistakes

• Confusing arithmetic (add) with geometric (multiply)
• Finding the nth term formula but not checking it with known terms
• Forgetting that n starts at 1, not 0
• Using the term-to-term rule to find a distant term (inefficient — use the nth term formula instead)
• Sign errors with negative common differences

Tips and Tricks

• Always check your nth term formula with at least two terms
• For arithmetic sequences: the nth term always has the form dn + c
• To find c quickly: work out d × 1, then see what you need to add/subtract to reach the first term
• If the differences are not constant, try looking at second differences — they reveal quadratic sequences (nth term involves n²)
• Geometric sequences with a ratio between 0 and 1 get smaller and approach zero