IGCSE Maths Sequences & nth Term Notes
A sequence is an ordered list of numbers following a rule. The nth term is a formula that generates any term from its position n.
Key concepts
- Linear (arithmetic) sequence
- Constant difference between terms; nth term is dn + c.
- Quadratic sequence
- Second difference is constant; nth term contains an n² term.
- Term-to-term vs position-to-term
- Term-to-term tells you the step; position-to-term (nth term) uses n directly.
Key formulas
Linear nth term
nth term = dn + (a − d)
d = common difference, a = first term.
Quadratic nth term
an² + bn + c
2a = second difference.
Worked example
Find the nth term of 5, 8, 11, 14, …
- 1The common difference is 3, so start with 3n.
- 23 × 1 = 3, but the first term is 5, so add 2.
- 3Write the nth term.
Answer: 3n + 2
Common mistakes
✗ Using term-to-term when the nth term is needed.
✓ The nth term must be a formula in n, not a step rule.
✗ Wrong constant in dn + c.
✓ Substitute n = 1 to check it gives the first term.
✗ Ignoring the second difference for quadratics.
✓ Half the second difference is the coefficient of n².
Quick check
Try these, then reveal the answers.
nth term of 2, 4, 6, 8?Reveal answer
2n
nth term of 7, 10, 13?Reveal answer
3n + 4
10th term of 3n − 1?Reveal answer
29
Exam tips
- State the common difference before writing dn.
- For quadratics, find the second difference first, then work out b and c.
- Always verify by substituting n = 1 and n = 2.
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Frequently asked questions
How do I find the nth term of a linear sequence?+
The coefficient of n is the common difference; add or subtract a constant so n = 1 gives the first term.
How do I know a sequence is quadratic?+
The first differences change but the second differences are constant — that constant equals 2a in an² + bn + c.
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