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WMA12/01 Jan 2024, Question 5: Geometric Progressions and Sum to Infinity

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

The question, in short

Find the sum of an infinite geometric series with first term 6 × 0.25^r, then analyse a recursively defined sequence u₁ = 3, uₙ₊₁ = (uₙ − 3)/(uₙ − 2) for periodicity and a 70-term sum.

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

Part (i) · 3 marks

Find the value of the infinite sum from r = 1 to ∞ of 6 × (0.25)ʳ.

  1. 1Geometric series: first term a = 6 × 0.25 = 1.5, common ratio r = 0.25
  2. 2|r| < 1, so S∞ = a/(1 − r)
  3. 3S∞ = 1.5/0.75 = 2

Answer2

Part (iia) · 2 marks

Show that the sequence defined by u₁ = 3 and uₙ₊₁ = (uₙ − 3)/(uₙ − 2) is periodic.

  1. 1u₂ = (3 − 3)/(3 − 2) = 0
  2. 2u₃ = (0 − 3)/(0 − 2) = 3/2
  3. 3u₄ = (3/2 − 3)/(3/2 − 2) = (−3/2)/(−1/2) = 3
  4. 4u₄ = u₁, so the sequence repeats: it is periodic

AnswerShown: u₁ = 3, u₂ = 0, u₃ = 3/2, u₄ = 3 = u₁, so the terms repeat and the sequence is periodic.

Part (iib) · 1 mark

State the order of this periodic sequence.

  1. 1The terms cycle 3, 0, 3/2, 3, 0, 3/2, …
  2. 2The cycle length is 3, so the order is 3

Answer3

Part (iic) · 2 marks

Hence find the sum from n = 1 to 70 of uₙ.

  1. 1Each cycle of 3 terms sums to 3 + 0 + 3/2 = 4.5
  2. 270 = 23 × 3 + 1, so there are 23 full cycles and one extra term u₇₀ = u₁ = 3
  3. 3Sum = 23 × 4.5 + 3 = 103.5 + 3 = 106.5

Answer106.5

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Also searched as: WMA12 January 2024 · WMA12/01 · wma12 jan24 · IAS Maths Pure Mathematics 2 January 2024 · question 5.
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