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4PM1/02 Jun 2020, Question 7: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 12 marks

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The question, in short

Geometric series with first three terms (x − 3), (x + 1), (4x − 2): find possible x, show convergence for x < 1, then find S and n from S/Sₙ = 256/255.

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Part (a) · 5 marks

Use the common ratio between consecutive terms to form an equation and solve for the two possible values of x.

  1. 1(x + 1)/(x − 3) = (4x − 2)/(x + 1)
  2. 2(x + 1)² = (4x − 2)(x − 3)
  3. 33x² − 16x + 5 = 0
  4. 4(x − 5)(3x − 1) = 0 ⇒ x = 5 or 1/3

Answerx = 5 or x = 1/3

Part (b) · 2 marks

Using x < 1, show the common ratio satisfies the condition for a convergent geometric series.

  1. 1x = 1/3: r = (x + 1)/(x − 3) = (4/3)/(−8/3)
  2. 2r = −1/2
  3. 3|r| < 1, so the series converges

Answerr = −1/2, and −1 < r < 1 so the series is convergent

Part (c) · 2 marks

Find the sum to infinity S using the first term and common ratio.

  1. 1a = x − 3 = −8/3
  2. 2S = a/(1 − r) = (−8/3)/(3/2)
  3. 3= −16/9

AnswerS = −16/9

Part (d) · 3 marks

Given S/Sₙ = 256/255, solve for the number of terms n.

  1. 1S/Sₙ = [a/(1 − r)] × [(1 − r)/(a(1 − rⁿ))] = 1/(1 − rⁿ)
  2. 21/(1 − rⁿ) = 256/255 ⇒ 255 = 256(1 − rⁿ)
  3. 3rⁿ = 1/256 ⇒ (−1/2)ⁿ = 1/256
  4. 4n = 8

Answern = 8

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