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0580/41 May/Jun 2025, Question 5: Quadratic and Cubic Sequences

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 6 marks

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The question, in short

Work with three sequences: identify a term from the first five given terms, derive a general formula for a second sequence, and use a sum formula for a third.

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

Part (a) · 1 mark

Determine which value occupies position 10 in the sequence 1, 8, 27, 64, 125.

  1. 1The terms are cube numbers 1³, 2³, 3³, 4³, 5³
  2. 210th term = 10³ = 1000

Answer1000 or 10³

Part (b) · 2 marks

Express the nth term of the sequence 5, 8, 13, 20, 29 as a formula in n.

  1. 1Differences: 3, 5, 7, 9; second difference = 2, so start with n²
  2. 2n²: 1, 4, 9, 16, 25; sequence − n² = 4 each time
  3. 3nth term = n² + 4

Answern² + 4

Part (ci) · 1 mark

Substitute n = 2 into n/2 (5n − 1) to calculate the total of the first two terms.

  1. 12/2 × (5 × 2 − 1) = 1 × 9 = 9

Answer9

Part (cii) · 2 marks

Using the sum formula n/2 (5n − 1), work out the value of the third term.

  1. 13rd term = (sum of first 3 terms) − (sum of first 2 terms)
  2. 2Sum of first 3 = 3/2 × (5 × 3 − 1) = 3/2 × 14 = 21
  3. 3Sum of first 2 = 2/2 × (5 × 2 − 1) = 9
  4. 43rd term = 21 − 9 = 12

Answer12

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