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0580/13 Oct/Nov 2025, Question 21: The nth Term of a Linear Sequence

Cambridge IGCSE Mathematics 0580 · Paper 13 (Core) · Non-calculator · 6 marks

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The question, in short

Sequence starts −6, 1, 8, 15. Find the next term, the term-to-term rule, the nth term, and decide whether 688 belongs to the sequence with reasoning.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 1 mark

Write down the term that follows 15 in the sequence.

  1. 1Differences: 1 − (−6) = 7, 8 − 1 = 7, 15 − 8 = 7
  2. 2Next term = 15 + 7 = 22

Answer22

Part (b) · 1 mark

State the rule linking each term to the one before it.

  1. 1Each term is 7 more than the one before: add 7

AnswerAdd 7

Part (ci) · 2 marks

Find an expression for the nth term of the sequence.

  1. 1Common difference 7, so nth term = 7n + c
  2. 2n = 1: 7 + c = −6, so c = −13
  3. 3nth term = 7n − 13

Answer7n − 13

Part (cii) · 2 marks

Decide whether 688 is in the sequence and explain your reasoning.

  1. 17n − 13 = 688
  2. 27n = 701
  3. 3n = 701 ÷ 7 = 100.14…, not a whole number
  4. 4No, 688 is not a term because 701 is not divisible by 7

AnswerNo, 7n − 13 = 688 gives 7n = 701 and 701 is not divisible by 7

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