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4PM1/02R Jun 2019, Question 7: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 11 marks

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The question, in short

Geometric series G has nth term uₙ, first term a, common ratio r > 0, with u₃ = 4 and u₇ = 16. Find r, a, a threshold n, and S₂₀ in a specified form.

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

Part (a) · 3 marks

Show that r = √2 using u₃ = 4 and u₇ = 16, then find the value of a.

  1. 1ar² = 4 and ar⁶ = 16
  2. 2u₇/u₃ = r⁴ = 16/4 = 4
  3. 3r² = 2 ⇒ r = √2 (r > 0)
  4. 4a = 4/(√2)² = 2

Answerr = √2 shown; a = 2

Part (b) · 4 marks

Find the smallest n for which uₙ exceeds 500.

  1. 12 × (√2)^(n − 1) > 500 ⇒ (√2)^(n − 1) > 250
  2. 2(n − 1) lg √2 > lg 250
  3. 3n − 1 > 15.93…
  4. 4n > 16.93… ⇒ n = 17

Answern = 17

Part (c) · 4 marks

Find S₂₀, giving the answer as p(1 + √2) with p an integer.

  1. 1S₂₀ = 2((√2)²⁰ − 1)/(√2 − 1)
  2. 2= 2046/(√2 − 1)
  3. 3Multiply by (√2 + 1)/(√2 + 1): 2046(√2 + 1)/(2 − 1)
  4. 4= 2046(1 + √2)

Answer2046(1 + √2)

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