What is a number sequence?
A number sequence is a list of numbers that follow each other while always obeying the same rule. Once you've spotted the rule, you can continue the sequence for as long as you like, or work out a missing term in the middle.
2, 5, 8, 11, ?
Adding 3 each time → the missing term is 14
Increasing or decreasing sequences
A sequence is increasing when the numbers go up at every step, and decreasing when they go down.
- Increasing: 4, 9, 14, 19, 24… (adding 5 each time)
- Decreasing: 30, 26, 22, 18, 14… (subtracting 4 each time)
Finding the rule of a sequence
The method is always the same, whatever the sequence: you work out the gap between two terms that follow each other, and check that the gap is the same everywhere.
- Take the first two terms and work out the gap between them (the second minus the first).
- Check that same gap works for the following terms too.
- Once the rule is confirmed, apply it to the last known term to find the next one.
Example: in 7, 13, 19, 25…, the gap is always +6 (13−7=6, 19−13=6, 25−19=6). So the next term is 25+6 = 31.
Arithmetic sequences
A sequence where you always add (or subtract) the same number is called an arithmetic sequence. It's the most common type from around age 9. This fixed number is called the sequence's common difference.
100, 91, 82, 73, 64…
Arithmetic sequence with a common difference of −9 (subtracting 9 each time)
Multiplicative sequences
In a multiplicative sequence, you always multiply by the same number to get from one term to the next, instead of adding. These sequences grow (or shrink) much faster than an arithmetic sequence.
3, 6, 12, 24, 48…
Each term is multiplied by 2
Counting in 2s, 5s and 10s
Counting in regular jumps is the very first arithmetic sequence a child comes across, from age 6: it's excellent preparation for times tables.
- In 2s: 0, 2, 4, 6, 8, 10…
- In 5s: 0, 5, 10, 15, 20…
- In 10s: 0, 10, 20, 30, 40…
Mistakes to avoid
- Only checking the gap between the first two terms, without confirming it across the whole sequence.
- Confusing an arithmetic sequence (adding) with a multiplicative sequence (multiplying): test both if addition doesn't "fit".
- Getting the direction wrong on a decreasing sequence and adding instead of subtracting.
Worked exercises
Continue the sequence: 15, 20, 25, 30, …
What is the rule of the sequence 50, 44, 38, 32?
Continue the multiplicative sequence: 2, 6, 18, 54, …
Now it's your turn!
On EasyMaths, your child practises continuing number sequences with exercises that never repeat and instant marking.
Start the exercises →Frequently asked questions
How do you find the rule of a number sequence?
You work out the gap between two consecutive terms (the second minus the first). If that gap is the same everywhere, that's the rule: you always add (or subtract) that same number to get from one term to the next.
What's the difference between an increasing and a decreasing sequence?
An increasing sequence goes up at every step (2, 5, 8, 11… adding 3 each time). A decreasing sequence goes down at every step (20, 17, 14, 11… subtracting 3 each time).
What is a multiplicative sequence?
Instead of always adding the same number, you always multiply by the same number to get from one term to the next. For example 2, 4, 8, 16, 32: each term is multiplied by 2.