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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.

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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, …

  1. 1The common difference is 3, so start with 3n.
  2. 23 × 1 = 3, but the first term is 5, so add 2.
  3. 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

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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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