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WMA12/01 Jan 2023, Question 10: Proof by Exhaustion and Counter-example

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 4 marks

The question, in short

Complete a proof by exhaustion that 2n² + n + 1 is never a multiple of 3 for integer n, following the given n = 3k case.

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

Solution · 4 marks

  1. 1Every integer is 3k, 3k + 1 or 3k + 2
  2. 2n = 3k + 1: 2(3k + 1)² + (3k + 1) + 1 = 18k² + 15k + 4
  3. 3= 3(6k² + 5k + 1) + 1, not divisible by 3
  4. 4n = 3k + 2: 2(3k + 2)² + (3k + 2) + 1 = 18k² + 27k + 11
  5. 5= 3(6k² + 9k + 3) + 2, not divisible by 3
  6. 6All three cases leave remainder 1 or 2, so 2n² + n + 1 is never divisible by 3

AnswerShown: n = 3k + 1 gives 3(6k² + 5k + 1) + 1 and n = 3k + 2 gives 3(6k² + 9k + 3) + 2, so with n = 3k no case is divisible by 3

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Also searched as: WMA12 January 2023 · WMA12/01 · wma12 jan23 · IAS Maths Pure Mathematics 2 January 2023 · question 10.
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