GCSE · Algebraic proof

Expressing generality — the language of proofExam-style questions

Why testing examples proves nothing, how 2n, 2n + 1 and n, n + 1, n + 2 encode "any even", "any odd" and "consecutive", and when one counterexample settles everything.

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

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
Prove that the sum of any two consecutive odd numbers is a multiple of 4.
Level 4 · 3 marksTry it
2
Tomas says, "The sum of any two prime numbers is always even."

Show that Tomas is wrong.
Level 4 · 2 marksTry it
3
Prove that $n^2 + n$ is even for every integer $n$ .
Level 4 · 3 marksTry it
4
Aisha says, "The difference between the squares of any two consecutive integers is always even."

Show that Aisha is wrong.
Level 4 · 2 marksTry it
5
Prove that the sum of any four consecutive odd numbers is a multiple of 8.
Level 4 · 4 marksTry it