GCSE Maths Sequences Notes
A sequence is an ordered list of numbers following a rule. The nth term formula lets you find any term without listing them all.
Key concepts
- Term-to-term rule
- How you get from one term to the next, e.g. '+4 each time'. Useful but slow for large positions.
- nth term (position-to-term)
- A formula in n that gives any term directly, e.g. 4n − 1 gives the 100th term instantly.
- Linear sequence
- A constant difference between terms. The nth term is (difference)n + (adjustment).
- Quadratic sequence
- The second differences are constant. The nth term contains an n² term.
Key formulas
nth term of a linear sequence
nth term = dn + (a − d)
d = common difference, a = first term.
Fibonacci-type rule
each term = sum of the two before it
Worked example
Find the nth term of 3, 7, 11, 15, …
- 1Find the common difference: 7 − 3 = 4, so d = 4.
- 2The nth term starts 4n.
- 3Check 4 × 1 = 4, but the first term is 3, so subtract 1.
- 4nth term = 4n − 1.
Answer: nth term = 4n − 1
Common mistakes
✗ Writing the term-to-term rule (+4) as the nth term.
✓ The nth term must be a formula in n, e.g. 4n − 1, not '+4'.
✗ Forgetting the adjustment number after dn.
✓ Compare dn with the actual first term and add or subtract to match.
✗ Assuming a sequence is linear without checking.
✓ If the first differences change, check the second differences — it may be quadratic.
Quick check
Try these, then reveal the answers.
Find the nth term of 5, 8, 11, 14, …Reveal answer
3n + 2
What is the 10th term of 4n − 1?Reveal answer
39
Is 2, 4, 8, 16 linear?Reveal answer
No — it doubles, so it is geometric.
Exam tips
- Always show the common difference — it earns a method mark on its own.
- To test if a number is in a sequence, set the nth term equal to it and solve for n; a whole number means yes.
- For quadratic sequences, halve the constant second difference to get the coefficient of n².
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Frequently asked questions
How do I find the nth term of a linear sequence?+
Find the common difference d, write dn, then add or subtract a number so it matches the first term.
How can I tell a sequence is quadratic?+
The differences between terms are not constant, but the differences of those differences (second differences) are.
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