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A-LevelFurther MathematicsProof by induction9231/12 · May/June 2012 · Q25 Marks

Prove, by mathematical induction, that, for integers n ≥ 2, 4ⁿ > 2n + 3ⁿ.

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

  1. [B1] (States proposition.) (Pₙ: 4ⁿ > 2ⁿ + 3ⁿ)
  2. [B1] Proves base case. Let n = 2, 16 > 4 + 9 ⇒ P₂ is true.
  3. [B1] States inductive hypothesis. Assume Pₖ is true ⇒ 4ᵏ > 2ᵏ + 3ᵏ
  4. [M1] Proves inductive step. 4ᵏ⁺¹ = 4.4ᵏ > 4(2ᵏ + 3ᵏ ) = 4.2ᵏ + 4.3ᵏ
  5. [A1] > 2.2ᵏ + 3.3ᵏ = 2ᵏ⁺¹ + 3ᵏ⁺¹
  6. [A1 (CWO)] States conclusion. Hence result true, by PMI, for all integers n ≥ 2.

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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, May/June 2012 Paper 1 Variant 2, question 2.

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