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A-LevelFurther MathematicsProof by induction9231/12 · Oct/Nov 2020 · Q25 Marks

Prove by mathematical induction that 7²ⁿ – 1 is divisible by 12 for every positive integer n.

  1. (a)Prove by mathematical induction that 7²ⁿ – 1 is divisible by 12 for every positive integer n.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

  1. [B1] 72-1=48 is divisible by 12. Checks base case.
  2. [B1] Assume that 72k −1 is divisible by 12 for some positive integer k. States inductive hypothesis.
  3. [M1] Then 72k+2-1=49.72k −1 = 48.72k +72 -1 Or (72k+2 −1)–(72k − 1) = 48.72k Separates 72k -1.
  4. [A1] 72k+2 −1 is divisible by 12.
  5. [A1] So if is true for n = k (could be written earlier) it is also true for n = k+1. Hence, by induction, true for every positive integer n. Depends on previous M1 and A1.

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About this question

Topic
Proof by induction, A-Level Further Mathematics (syllabus 9231). Structured question, 5 marks.
Source
A-Level Further Mathematics, Oct/Nov 2020 Paper 1 Variant 2, question 2.

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