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4PM1/02 Jun 2021, Question 5: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 7 marks

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The question, in short

Geometric series with nth term uₙ, common ratio r, given u₂ + u₄ = 212.5 and u₃ + u₄ = 62.5; find possible r and then exact S for the convergent series.

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Part (a) · 5 marks

Use the two given sums of terms to form equations and solve for the two possible values of the common ratio r.

  1. 1ar + ar³ = 212.5 and ar² + ar³ = 62.5
  2. 2Divide: (1 + r²)/(r + r²) = 212.5/62.5 = 17/5
  3. 35 + 5r² = 17r + 17r² ⇒ 12r² + 17r − 5 = 0
  4. 4(4r − 1)(3r + 5) = 0
  5. 5r = 1/4 or r = −5/3

Answerr = 1/4 or r = −5/3

Part (b) · 2 marks

Using the convergent case, find the exact sum to infinity S of the series.

  1. 1Convergent so r = 1/4
  2. 2a(1/4 + 1/64) = 212.5 ⇒ a = 800
  3. 3S = a/(1 − r) = 800/(3/4) = 3200/3

AnswerS = 3200/3

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