Number sequences

Numbers in a line

A sequence is a list of numbers in order that follows a rule. Each number is a term, and its position is its term number.

The rule here is "start at 3 and add 4 each time". The 6th term would be 23.

Finding the rule

1. Look at how each term changes into the next.
2. Write the change between each pair.
3. Check the change is the same all the way along (or follows its own pattern).

Example: 50, 44, 38, 32 … The difference is −6 each time. Rule: "start at 50 and subtract 6".

Types of number sequences

Increasing and decreasing

• Increasing sequences get bigger: 4, 8, 12, 16.
• Decreasing sequences get smaller: 100, 90, 80, 70.

Famous sequences

• Even numbers: 2, 4, 6, 8 …
• Odd numbers: 1, 3, 5, 7 …
• Triangular numbers: 1, 3, 6, 10, 15 … (the dots needed to make bigger and bigger triangles)
• Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13 … Each term is the sum of the two before it. It shows up in pinecones and sunflower seed spirals.

Finding a missing term

Example: 6, 13, __, 27, 34

The difference is 7 (13 − 6 = 7, 34 − 27 = 7). So the missing term is 13 + 7 = 20.

Worked examples

Example 1: Find the next two terms: 5, 10, 20, 40, __, __

Each term doubles: 80, 160.

Example 2: What is the rule for 81, 72, 63, 54?

Start at 81 and subtract 9.

Example 3: What is the 10th term of 3, 6, 9, 12 …?

The terms are 3 × the term number: 3 × 10 = 30.

Common mistakes

• Only checking the first two terms. Make sure the rule works for all of them.
• Assuming every sequence adds the same amount. Some multiply or have growing differences.
• Mixing up term and term number.

Try it yourself

1. Next term: 7, 14, 21, 28, __
2. Rule for 2, 6, 18, 54?
3. Missing term: 100, __, 60, 40
4. Next term: 1, 3, 6, 10, __

Answers: 1) 35 2) start at 2 and multiply by 3 3) 80 4) 15

Quick recap

• A sequence follows a rule from term to term.
• Find the difference (or multiplier) between terms.
• Sequences can be increasing or decreasing.