GCSE · Algebraic proof

Proofs with squaresExam-style questions

The square-shaped proof toolkit: expanding (an + b)² honestly, reading parity out of 2(…) + 1, difference-of-two-squares shortcuts, and the completed square as an always-positive argument.

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13 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 3Level 4Level 5
  1. 1
    The nnth triangular number is n(n+1)2\dfrac{n(n + 1)}{2}

    Prove that 8 times any triangular number, plus 1, is always a square number.

    (3 marks)
    Level 4 · 3 marksTry it
  2. 2
    (a) Factorise x2−4y2x^2 - 4y^2

    (1 mark)
    (b) Show that 38−43^8 - 4 is not a prime number.

    (2 marks)
    Level 4 · 3 marksTry it
  3. 3
    Prove that (n+3)2−(n−3)2(n + 3)^2 - (n - 3)^2 is a multiple of 12 for every integer nn.
    Level 4 · 3 marksTry it
  4. 4
    Prove that the square of any odd number is odd.
    Level 4 · 3 marksTry it
  5. 5
    Prove that (3n+1)2−1(3n + 1)^2 - 1 is a multiple of 3 for every integer nn.
    Level 4 · 3 marksTry it

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