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WMA12/01 Jun 2022, Question 6: Geometric Progressions and Sum to Infinity

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Geometric sequence with common ratio r, u₂ + u₃ = 6 and u₄ = 8: show r satisfies 3r² − 4r − 4 = 0, then find u₁ and S∞ given a sum to infinity exists.

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

Part (a) · 3 marks

Using u₂ + u₃ = 6 and u₄ = 8, derive and show the quadratic equation 3r² − 4r − 4 = 0 for the common ratio r.

  1. 1Using uₙ = arⁿ⁻¹: ar + ar² = 6 and ar³ = 8
  2. 2Divide: (ar + ar²)/(ar³) = 6/8, so (1 + r)/r² = 3/4
  3. 34 + 4r = 3r²
  4. 43r² − 4r − 4 = 0

AnswerShown: (ar + ar²)/ar³ = 6/8 gives 1 + r = (3/4)r², so 3r² − 4r − 4 = 0

Part (b) · 3 marks

Given the sequence has a sum to infinity, find the first term u₁.

  1. 13r² − 4r − 4 = 0 gives (3r + 2)(r − 2) = 0, so r = −2/3 or r = 2
  2. 2A sum to infinity needs |r| < 1, so r = −2/3
  3. 3ar³ = 8, so a = 8/(−2/3)³ = 8/(−8/27)
  4. 4u₁ = −27

Answeru₁ = −27

Part (c) · 2 marks

Find S∞, the sum to infinity of the sequence.

  1. 1S∞ = a/(1 − r) = −27/(1 − (−2/3))
  2. 2= −27/(5/3) = −81/5

AnswerS∞ = −81/5

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