A-LevelFurther MathematicsProof by induction9231/12 · May/June 2014 · Q36 Marks

Prove by mathematical induction that, for all non-negative integers n, 11^(2n) + 25^n + 22 is divisible by 24.

Prove by mathematical induction that, for all non-negative integers n, 11^(2n) + 25^n + 22 is divisible by 24.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.

[B1] H_n: f(k) = 11^(2k) + 25^k + 22 = 24λ
[B1] f(0) = 1 + 1 + 22 = 24 = 1 × 24 ⇒ H_0 is true.
[M1] f(k + 1) − f(k) = 11^(2k+2) + 25^(k+1) + 22 – (11^(2k) + 25^k + 22)
[A1] = 11^(2k)(121 – 1) + 25^k(25 – 1)
[A1] = 11^(2k) × 24 × 5 + 25^k × 24 = 24μ
[(M1)] f(k + 1) = 11^(2k+2) + 25^(k+1) + 22
[(A1)] = 121.11^(2k) + 25.25^k + 22 = (120 + 1)11^(2k) + (24 + 1)25^k + 22 (OE)
[(A1)] = 120.11^(2k) + 24.25^k + 24λ = 24μ
[A1] ⇒ f(k + 1) = 24μ + 24λ = 24(μ + λ) ⇒ H_k + 1 is true. Hence by PMI H_n is true for all non-negative integers. (Must see non-negative integers.) CSO: Final mark requires all previous marks.

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Topic: Proof by induction, A-Level Further Mathematics (syllabus 9231). Structured question, 6 marks.
Source: A-Level Further Mathematics, May/June 2014 Paper 1 Variant 2, question 3.

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