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4PM1/02R Jun 2024, Question 8: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 14 marks

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The question, in short

Geometric series G has first two terms summing to 360 and second plus third terms summing to 288. Find the nth term form, convergence, sum to infinity, and least terms for a given sum.

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

Part (a) · 7 marks

Show the nth term equals A(4/5)ⁿ⁻¹ and find the integer A.

  1. 1a + ar = 360 and ar + ar² = 288
  2. 2r = ar(1 + r)/(a(1 + r)) = 288/360 = 4/5
  3. 3a = 360/(1 + 4/5) = 200
  4. 4Uₙ = 200(4/5)ⁿ⁻¹

AnswerUₙ = 200(4/5)ⁿ⁻¹ (A = 200)

Part (b) · 1 mark

Explain why the geometric series G converges.

  1. 1r = 4/5
  2. 2|r| < 1, so the series converges

Answer|r| = 4/5 < 1, so G is convergent

Part (c) · 2 marks

Use the common ratio to find the sum to infinity of G.

  1. 1S∞ = a/(1 − r)
  2. 2= 200/(1 − 4/5) = 1000

Answer1000

Part (d) · 4 marks

Find the smallest number of terms whose sum exceeds 978.

  1. 1200(1 − (4/5)ⁿ)/(1 − 4/5) > 978
  2. 21000(1 − (4/5)ⁿ) > 978, so (4/5)ⁿ < 11/500
  3. 3n log(4/5) < log(11/500)
  4. 4n > 17.10…, so n = 18

Answern = 18

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