A-level · Proof

Proof by deductionExam-style questions

Direct proof — starting from things known to be true and reasoning in valid algebraic steps to the conclusion. Representing the general case (2n, 2n+1, consecutive integers), "show that" manipulations, completing the square for positivity, and always finishing with a clear conclusion.

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12 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 3Level 4Level 5
  1. 1
    Prove that the sum of the squares of any two consecutive odd integers is even but not divisible by 44. (4 marks)
    Level 3 · 4 marksTry it
  2. 2
    (a) Prove that the sum of any three consecutive integers is divisible by 33. (2 marks)
    (b) Show, by means of a counter-example, that the sum of any four consecutive integers need not be divisible by 44. (2 marks)
    Level 3 · 4 marksTry it
  3. 3
    Prove that n3−nn^3 - n is divisible by 66 for all positive integers nn. (4 marks)
    Level 3 · 4 marksTry it
  4. 4
    Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 88. (4 marks)
    Level 3 · 4 marksTry it
  5. 5
    Prove, using algebra, that

    (2n+5)2−(2n−1)2(2n + 5)^2 - (2n - 1)^2

    is a multiple of 24 for all n∈Nn \in \mathbb{N}

    (3 marks)
    Level 3 · 3 marksTry it

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