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0606/23 May/Jun 2026, Question 11: Geometric Progressions and Sum to Infinity

Cambridge IGCSE Additional Mathematics 0606 · Paper 23 (Calculator) · Calculator · 8 marks

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The question, in short

Use geometric progression terms with first term a and ratio r to derive a quadratic in r, then solve for r and an integer k.

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

Part (a) · 4 marks

Show that the given relation between consecutive and skip terms leads to r² + r − 6 = 0.

  1. 1u_n = ar^(n−1)
  2. 2arⁿ − ar^(n−1) = (1/6)(ar^(n+2) − arⁿ)
  3. 3Divide by ar^(n−1): r − 1 = (1/6)(r³ − r) = (1/6)r(r − 1)(r + 1)
  4. 4r ≠ 1, so 6 = r(r + 1)
  5. 5r² + r − 6 = 0

Answerr² + r − 6 = 0 shown

Part (b) · 4 marks

Given u_(k+2) − u_k = 48a, find the ratio r and the integer k.

  1. 1(r + 3)(r − 2) = 0, r > 0 ⇒ r = 2
  2. 2u_(k+2) − u_k = a × 2^(k+1) − a × 2^(k−1) = 3a × 2^(k−1)
  3. 33a × 2^(k−1) = 48a ⇒ 2^(k−1) = 16
  4. 4k − 1 = 4 ⇒ k = 5

Answerr = 2, k = 5

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Also searched as: 0606/23/M/J/26 · 0606_s26_qp_23 · 0606 s26 23 · question 11.
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