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0606/11 May/Jun 2026, Question 9: Geometric Progressions and Sum to Infinity

Cambridge IGCSE Additional Mathematics 0606 · Paper 11 (Non-calculator) · Non-calculator · 8 marks

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The question, in short

Geometric progression with first term a and ratio r: use the 4th and 7th term relationship, then the sum of the first 5 terms, and a trigonometric GP to find its infinite sum.

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

Part (ai) · 2 marks

Given the 4th term equals −8 times the 7th term, find the common ratio r.

  1. 1ar³ = −8ar⁶
  2. 2r³ = −1/8
  3. 3r = −1/2

Answerr = −1/2

Part (aii) · 2 marks

Given the sum of the first 5 terms is 99/32, find the first term a.

  1. 1S₅ = a(1 − (−1/2)⁵)/(1 + 1/2) = a(33/32)/(3/2) = 11a/16
  2. 211a/16 = 99/32
  3. 3a = 9/2

Answera = 9/2

Part (b) · 4 marks

For the GP with first terms cos θ sin θ, cos θ sin³θ, cos θ sin⁵θ, show the sum to infinity equals tan θ.

  1. 1r = cos θ sin³θ/(cos θ sin θ) = sin²θ, and 0 < sin²θ < 1
  2. 2S∞ = cos θ sin θ/(1 − sin²θ)
  3. 3= cos θ sin θ/cos²θ
  4. 4= sin θ/cos θ = tan θ

AnswerS∞ = tan θ shown

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