A-level · Beta

Proof and counterexamples

Build a deduction, repair a false proof and test a universal claim.

n(n+1)n(n+1)

Prove the product is always even

    Prove n(n + 1) is even for every integer n. Which opening covers every case?

    Choose a justified step. An incorrect choice gives feedback without advancing the argument.

    Predict, test and explain

    Step 1 of 3

    Predict

    Do many confirming examples prove a claim about all integers?

    InvestigateWhy is a counterexample enough to disprove a universal statement?

    Explore and compare

    Choose the investigation above. Predict what will change before you move a control.

    Try it in the simulation. Change one thing at a time.

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    Hints, self-check and connections
    Try a self-check

    Fixed values, separate from the diagram controls.

    Do many confirming examples prove a claim about all integers?

    n(n+1)n(n+1)

    Hint

    State the domain and distinguish evidence from a general proof.

    Reveal answer

    A single counterexample disproves a universal claim. Confirming examples alone do not prove infinitely many cases.

    2∣n(n+1)2\mid n(n+1)

    Build an explanation

    Hint 1

    State the domain and distinguish evidence from a general proof.

    Hint 2

    A single counterexample disproves a universal claim. Confirming examples alone do not prove infinitely many cases.

    Hint 3

    Explain why the rule is valid, including its assumptions.

    Worked example

    This example uses fixed values, separate from the diagram controls.

    Prove the product of consecutive integers is even.

    1. Every integer is even or odd.
    2. If n is odd, n + 1 is even; if n is even, the first factor is already even.
    3. Either way the product has a factor of 2.

    Answer: n(n + 1) is even for every integer n.

    Connect this idea

    Watch out: A single counterexample disproves a universal claim. Confirming examples alone do not prove infinitely many cases.

    Specification and learning route

    AQA A1 · Edexcel Pure 1.1

    AS and A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

    Useful starting knowledge: Integer arithmetic; Factors; Logical statements.

    AQA specification · Edexcel specification