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4PM1/01 Jun 2024, Question 8: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 12 marks

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The question, in short

Geometric series G has first plus second term 400 and second plus third term 100. Show the common ratio and first term, find the infinite sum, and use logs to find the smallest n with Sₙ > 426.6.

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Part (a) · 4 marks

Using the two given pair sums, prove that the ratio between consecutive terms equals 1/4.

  1. 1a + ar = 400
  2. 2ar + ar² = 100
  3. 3r(a + ar) = 100, so 400r = 100
  4. 4r = 1/4

Answerr = 1/4 shown

Part (b) · 2 marks

Prove that the series begins with 320.

  1. 1a(1 + r) = 400
  2. 2a = 400/(1 + 1/4) = 320

Answera = 320 shown

Part (c) · 2 marks

Calculate the total of all terms added forever.

  1. 1S∞ = a/(1 − r)
  2. 2= 320/(1 − 1/4) = 1280/3

Answer1280/3

Part (d) · 4 marks

By taking logarithms, determine the smallest whole n for which the partial sum exceeds 426.6.

  1. 1320(1 − (1/4)ⁿ)/(1 − 1/4) > 426.6
  2. 2(1/4)ⁿ < 1/6400, i.e. 4ⁿ > 6400
  3. 3n > log 6400/log 4 = 6.32…
  4. 4Least n = 7

Answern = 7

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