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4PM1/02 Jan 2023, Question 6: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 8 marks

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The question, in short

Geometric series G has nth term Uₙ with first three terms q(4p + 1), q(2p + 3), q(2p − 3). Find possible p, then given convergence with sum to infinity 250, find q.

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Part (a) · 5 marks

Use the geometric relationship between the first three terms to find the possible values of p.

  1. 1r = U₂/U₁ = U₃/U₂
  2. 2(2p + 3)/(4p + 1) = (2p − 3)/(2p + 3)
  3. 3(2p + 3)² = (4p + 1)(2p − 3)
  4. 44p² + 12p + 9 = 8p² − 10p − 3
  5. 52p² − 11p − 6 = 0, (2p + 1)(p − 6) = 0
  6. 6p = −1/2 or p = 6

Answerp = −1/2 or p = 6

Part (b) · 3 marks

Given G is convergent with sum to infinity 250, find the value of q.

  1. 1p = 6: r = 9/15 = 3/5 (|r| < 1), U₁ = 25q
  2. 2(p = −1/2 gives r = −2, not convergent)
  3. 325q/(1 − 3/5) = 250
  4. 462.5q = 250, q = 4

Answerq = 4

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