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A-LevelFurther MathematicsProof by induction9231/12 · May/June 2024 · Q26 Marks

Prove by mathematical induction that 6⁴ⁿ + 38ⁿ - 2 is divisible by 74 for all positive integers n.

  1. Prove by mathematical induction that 6⁴ⁿ + 38ⁿ - 2 is divisible by 74 for all positive integers n.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

  1. [B1] 6⁴ +38¹ – 2 = 1332 is divisible by 74. Checks base case.
  2. [B1] Assume that 6⁴ᵏ + 38ᵏ – 2 is divisible by 74 for some positive integer k. States inductive hypothesis.
  3. [M1 A1] Then 6⁴ᵏ⁺⁴ +38ᵏ⁺¹ − 2 = (1295+1)×6⁴ᵏ +(37+1)×38ᵏ -2 Separates 6⁴ᵏ +38ᵏ – 2 or considers difference.
  4. [A1] Is divisible by 74 because 1295×6⁴ᵏ +37×38ᵏ is divisible by 74.
  5. [A1] Hence, by induction, 6⁴ᵏ +38ᵏ – 2 is divisible by 74, is 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 2024 Paper 1 Variant 2, question 2.

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