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WMA12/01 Oct 2024, Question 2: Sequences: Formulas and Recurrence

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

A sequence has u₁ = 7 and uₙ₊₁ = (−1)ⁿ uₙ + k with constant k. Show u₅ = 7, then use the sum of the first 4 terms being 30 to find k and the sum to 150 terms.

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

Part (a) · 3 marks

Using the recurrence with u₁ = 7, work out successive terms to show that u₅ equals 7.

  1. 1u₂ = (−1)¹ × 7 + k = k − 7
  2. 2u₃ = (−1)² (k − 7) + k = 2k − 7
  3. 3u₄ = (−1)³ (2k − 7) + k = 7 − k
  4. 4u₅ = (−1)⁴ (7 − k) + k = 7

AnswerShown: u₂ = k − 7, u₃ = 2k − 7, u₄ = 7 − k, so u₅ = k + (7 − k) = 7

Part (b) · 2 marks

Given that the sum of u₁ to u₄ is 30, find the constant k.

  1. 17 + (k − 7) + (2k − 7) + (7 − k) = 30
  2. 22k = 30
  3. 3k = 15

Answerk = 15

Part (c) · 2 marks

Using your value of k, find the sum of uᵣ from r = 1 to 150.

  1. 1The sequence repeats every 4 terms: 7, 8, 23, −8 (sum 30)
  2. 2150 = 37 × 4 + 2
  3. 3Sum = 37 × 30 + u₁ + u₂ = 1110 + 7 + 8
  4. 4= 1125

Answer1125

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Also searched as: WMA12 October 2024 · WMA12/01 · wma12 oct24 · IAS Maths Pure Mathematics 2 October 2024 · question 2.
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