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A-LevelFurther MathematicsProof by induction9231/12 · May/June 2025 · Q37 Marks

The sequence u₁, u₂, u₃, ... is such that u₁ = 5 and u_{n+1} = 6u_n + 5 for n ≥ 1.

  1. (a)Prove by induction that u_n = 6^n - 1 for all positive integers n.[5]
  2. (b)Deduce that u_{2n} is divisible by u_n for n ≥ 1.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

  1. [B1] Shows base case.
  2. [B1] States inductive hypothesis.
  3. [M1 A1] Substitutes into recursion formula.
  4. [A1] States conclusion.
  5. [M1 A1] Uses (a) and difference of two squares.

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About this question

Topic
Proof by induction, A-Level Further Mathematics (syllabus 9231). Structured question, 7 marks.
Source
A-Level Further Mathematics, May/June 2025 Paper 1 Variant 2, question 3.

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