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0606/12 May/Jun 2026, Question 10: Geometric Progressions and Sum to Infinity

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 8 marks

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The question, in short

A geometric progression has 2nd, 3rd and 4th terms 11x − 1, x + 5 and x − 1, with x > 0. Find its common ratio and first term, then the integer k where the infinite sum equals k/3.

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Part (a) · 6 marks

Using the three given consecutive terms and x > 0, determine the progression's common ratio and its first term.

  1. 1(x + 5)/(11x − 1) = (x − 1)/(x + 5)
  2. 2x² + 10x + 25 = 11x² − 12x + 1
  3. 310x² − 22x − 24 = 0, i.e. 5x² − 11x − 12 = 0
  4. 4(5x + 4)(x − 3) = 0, so x = 3 (x > 0)
  5. 5Terms are 32, 8, 2, so r = 1/4
  6. 6First term = 32 ÷ 1/4 = 128

Answerr = 1/4, first term 128

Part (b) · 2 marks

With the infinite sum written as k/3, work out the integer k.

  1. 1S∞ = 128/(1 − 1/4) = 512/3
  2. 2k = 512

Answerk = 512

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