Number Sequences
💡Look at the gaps between terms first. A constant gap means you add the same amount each time to reach the next number.
Try it — find the next term
liveA constant gap means you add the same amount each time.
- 1
Look at the gaps
common difference = 5
- 2
Add it to the last term
22 + 5 = 27
⏱️ Practice against the clock
A worked example
7, 12, 17, 22, __: each step adds 5, so the next term is 27.
Always look at the gaps first
First thing on any sequence: check the space between each pair of terms. If the gap stays the same, it is arithmetic and you are just adding a fixed amount each step. 7, 12, 17, 22 climbs by 5, so 27 comes next.
Arithmetic patterns turn up far more than any other kind, so this one look cracks most of what you will run into.
When the gaps will not sit still
If the differences keep changing, try two more things. Divide each term by the one before it. A steady result there means it is geometric. Or take the differences of the differences, and if those hold constant, you are looking at something quadratic.
Peel through the layers in that order, gaps then ratios then second differences, and almost any sequence gives itself up.
Questions people ask
What is the first thing to check in any sequence? ▾
The differences between neighbouring terms. A constant difference points to an arithmetic sequence, which is the most common pattern by far.
What if the differences are not constant? ▾
Check whether each term is a fixed multiple of the last (a geometric pattern), or look at the differences of the differences for something quadratic.