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0606/13 Oct/Nov 2025, Question 10: Geometric Progressions and Sum to Infinity

Cambridge IGCSE Additional Mathematics 0606 · Paper 13 (Non-calculator) · Non-calculator · 8 marks

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The question, in short

For an arithmetic series with first term a and difference d, use S20 = 3 × S10 to express a using d; then for a geometric series A with |r| < 1, relate the infinite sum of the even-term series B to that of A.

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

Part (a) · 3 marks

Using the condition linking the 20-term and 10-term sums of the arithmetic progression, determine a as a multiple of d.

  1. 1(20/2)(2a + 19d) = 3 × (10/2)(2a + 9d)
  2. 220a + 190d = 30a + 135d
  3. 310a = 55d
  4. 4a = (11/2)d

Answera = (11/2)d

Part (b) · 5 marks

Given B formed from even-numbered terms of A, find the ratio of B's infinite sum to A's infinite sum, simplified in r.

  1. 1S_A = a₁/(1 − r)
  2. 2B has first term a₂ = a₁r and common ratio r²
  3. 3S_B = a₁r/(1 − r²)
  4. 4S_B/S_A = (a₁r/(1 − r²)) × ((1 − r)/a₁)
  5. 5= r(1 − r)/((1 − r)(1 + r)) = r/(1 + r)

AnswerS_B/S_A = r/(1 + r)

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