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Algebra · Foundation & Higher

Sequences & nth term

A sequence is a list of numbers following a pattern. The nth term formula lets you find any term without listing them all.

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Key facts to remember

  • 1An arithmetic (linear) sequence has a constant difference between consecutive terms.
  • 2The nth term of an arithmetic sequence = dn + (a − d), where d = common difference and a = first term.
  • 3A geometric sequence has a constant ratio (multiply by the same number each time).
  • 4Quadratic sequences have a constant second difference.
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Formulas

nth term (arithmetic)
nth term = dn + (a − d)

d = common difference, a = first term

Alternatively
nth term = a + (n−1)d
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Worked examples

Example 1

Find the nth term of: 3, 7, 11, 15, ...

Working

  1. Common difference d = 7 − 3 = 4
  2. First term a = 3
  3. nth term = 4n + (3 − 4) = 4n − 1
  4. Check: n=1 → 4(1)−1 = 3 ✓, n=2 → 4(2)−1 = 7 ✓
Answernth term = 4n − 1
Example 2

Is 95 a term in the sequence 4n − 1?

Working

  1. Set 4n − 1 = 95
  2. 4n = 96
  3. n = 24
  4. 24 is a positive integer, so yes — it is the 24th term.
AnswerYes, 95 is the 24th term.
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Common mistakes

✗Using the wrong difference (e.g. using the first term instead of the common difference as the coefficient).
✗Getting the constant term wrong — always double-check with n = 1.
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Exam tips

✓Always verify your nth term formula by substituting n = 1 and n = 2.
✓To check if a value is in a sequence, set the formula equal to the value and check if n is a positive integer.

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