Number Sequences and Rules

What you will learn

• Say what a sequence and a term are.
• Find a rule that adds, subtracts or multiplies and use it to continue a sequence.
• Use a rule based on the position to find any term.

Before you start: you should be able to count in steps and know patterns. See Skip Counting in 2s, 5s and 10s and Repeating Patterns.

The idea

A sequence is a list of numbers in a set order that follows a rule. Each number in the list is called a term. The first term is the first number, the second term is the next, and so on. Spotting the rule lets you continue the sequence.

Adding or subtracting the same amount

Look at the gaps between neighbouring terms. If they are all the same, you have an adding rule. The gap is called the common difference.

A sequence can also go down. $50, 45, 40, 35$ has the rule “subtract $5$”.

Multiplying by the same number

If each term is the previous one times the same number, you have a multiplying rule. The numbers grow quickly.

A rule from the position

Instead of building each term from the last, a rule can use the position. For the sequence $3, 7, 11, 15, \dots$, term $n$ is $4n - 1$: for $n = 1$ you get $3$, for $n = 2$ you get $7$, and for $n = 10$ you get $39$. This lets you find the $100$th term without writing out the first $99$.

Patterns that are not simple

Not every list has a simple rule. $1, 4, 9, 16, 25$ is the square numbers: the gaps are $3, 5, 7, 9$, and they themselves form a pattern. When the gaps change, look at how they change.

Worked examples

Example 1: Continue a sequence

Find the next two terms of $7, 12, 17, 22, \dots$.

1. The gaps are $12 - 7 = 5$, $17 - 12 = 5$, $22 - 17 = 5$.
2. The rule is add $5$.
3. $22 + 5 = 27$ and $27 + 5 = 32$.

Answer: $27$ and $32$.

Example 2: A missing term

Find the missing term: $5, 10, \square, 40, 80$.

1. $10 \div 5 = 2$ and $80 \div 40 = 2$. The rule is multiply by $2$.
2. $10 \times 2 = 20$.
3. Check: $20 \times 2 = 40$.

Answer: $20$.

Example 3: Using the position rule

A sequence has the rule “term $n$ is $3n + 2$”. Find the $10$th term.

1. Put $n = 10$ into the rule.
2. $3 \times 10 + 2 = 32$.

Answer: $32$.

Check the rule with every pair of neighbours, not only the first. $2, 4, 7$ starts by adding $2$ but then adds $3$.

Why it matters

Sequences model saving a fixed amount each week, a population that doubles, or the steps in a pattern of tiles. The idea of a rule that uses the position, $n$, is the first step into algebra and functions.