How to answer "Prove" questions in GCSE Maths

What it means

A "prove" question demands a watertight argument that holds for every case. Substituting numbers and showing the statement works for one or two values is NOT a proof — it is an example. The argument must use letters, not numbers.

What examiners want

• Define the general form using letters (e.g. let n be any integer; let 2n be any even number)
• Manipulate the algebraic expression symbolically until you reach the required form
• End with a clear concluding sentence connecting the algebra back to the claim
• Use definitions correctly: "2n + 1" is odd; "2n" is even; "3n" is a multiple of 3, etc.

Worked example

Prove that the sum of three consecutive integers is always a multiple of 3.

Let the three consecutive integers be n, n + 1 and n + 2. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). Since 3(n + 1) is 3 multiplied by an integer, the sum is always a multiple of 3.

Common mistakes

• Trying numerical examples (e.g. "1 + 2 + 3 = 6 = 3 × 2") — this is verification, not proof
• Not stating what your letters represent at the start
• Forgetting the concluding sentence — even with the algebra right, you lose the final mark
• Confusing "show that" with "prove" — they are different question types

Marks tip

Always finish with a sentence that links your algebra back to the original claim. "Therefore the sum is a multiple of 3" is worth a mark on its own.

Related command words