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4PM1/02 Jan 2020, Question 7: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 12 marks

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The question, in short

Geometric series: 7th term 192, 8th term 1152. Find 4th term as a simplified fraction; for series G with t₃ = 24 and t₂ + t₃ + t₄ = −36, derive 2r² + 5r + 2 = 0, then find S given convergence.

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Part (a) · 3 marks

Work out the 4th term of the series as a fraction in simplest form.

  1. 1r = 1152/192 = 6
  2. 24th term = 192/6³ = 192/216
  3. 3= 8/9

Answer8/9

Part (b) · 5 marks

Using t₃ = 24 and t₂ + t₃ + t₄ = −36, show that r satisfies 2r² + 5r + 2 = 0.

  1. 1t₂ = 24/r and t₄ = 24r
  2. 224/r + 24 + 24r = −36
  3. 324 + 24r + 24r² = −36r
  4. 424r² + 60r + 24 = 0
  5. 5Divide by 12: 2r² + 5r + 2 = 0

Answer2r² + 5r + 2 = 0 shown

Part (c) · 4 marks

Given G converges, calculate its sum to infinity S.

  1. 1(2r + 1)(r + 2) = 0, so r = −1/2 or r = −2
  2. 2Convergent, so |r| < 1: r = −1/2
  3. 3a = t₃/r² = 24/(1/4) = 96
  4. 4S = 96/(1 − (−1/2)) = 64

AnswerS = 64

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