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4PM1/02R Jun 2022, Question 3: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 6 marks

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The question, in short

A geometric series with r > 0 has 3rd term 5 and 5th term 5/2. Find r, a, and the sum to infinity in the form p + q√2.

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

Part (a) · 4 marks

Find the exact value of r, then the value of a, using the given 3rd and 5th terms.

  1. 1ar² = 5 and ar⁴ = 5/2
  2. 2Divide: r² = (5/2)/5 = 1/2
  3. 3r > 0, so r = 1/√2 = √2/2
  4. 4a = 5/r² = 5 ÷ (1/2) = 10

Answer(i) r = √2/2 (ii) a = 10

Part (b) · 2 marks

Find the sum to infinity of the series, giving it as p + q√2 with integers p and q.

  1. 1S∞ = a/(1 − r) = 10/(1 − √2/2)
  2. 2= 20/(2 − √2)
  3. 3= 20(2 + √2)/((2 − √2)(2 + √2)) = 20(2 + √2)/2
  4. 4= 20 + 10√2

Answer20 + 10√2

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