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A-LevelFurther MathematicsProof by induction9231/12 · May/June 2021 · Q16 Marks

Prove by mathematical induction that 2⁴ⁿ + 31ⁿ – 2 is divisible by 15 for all positive integers n.

  1. Prove by mathematical induction that 2⁴ⁿ + 31ⁿ – 2 is divisible by 15 for all positive integers n.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

  1. [B1] Checks base case.
  2. [B1] States inductive hypothesis.
  3. [M1 A1] Separates 2⁴ᵏ +31ᵏ – 2 or considers difference.
  4. [A1] is divisible by 15 because 15×2⁴ᵏ +30×31ᵏ is divisible by 15.
  5. [A1] Hence, by induction, true for every positive integer n.

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About this question

Topic
Proof by induction, A-Level Further Mathematics (syllabus 9231). Structured question, 6 marks.
Source
A-Level Further Mathematics, May/June 2021 Paper 1 Variant 2, question 1.

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