Fiducia TogetherProof
Showing something is always true, not just true for the numbers you tried.
Please note: This page explains a word or a process in education. It is general information about the system, the law and the guidance in England — not advice about your child, your school or your exam. If a decision is going against you, get advice from one of the free services we link to before you agree to anything or miss a deadline. If you are revising, check your own exam board's specification: boards word things differently and yours is the one that marks you.
What it is
A proof is a logical argument establishing a general result. Algebraic proof uses letters to represent any number; geometric proof uses stated theorems.
Where it applies
Higher tier mostly, and "show that" questions at both tiers.
How it works
The standard algebraic representations are worth memorising: 2n for any even number, 2n + 1 for odd, consecutive integers as n, n+1, n+2. Then manipulate and show the required property falls out.
A counter-example disproves a statement. One is enough.
How to use it in your work
Testing three examples is not a proof and never scores as one.
Finish with a sentence saying what you have shown, in words. "Since 2(2n² + 2n + 1) is 2 × an integer, the result is always even."
Where to get help
Your teacher, your board's specification for what is on your paper, and Bitesize for worked examples. The maths song runs the topics in order.
Revise this with a song
These are ours. We write revision songs because a lot of people remember a tune when they cannot remember a page, and because revising out loud in the kitchen beats staring at a highlighter.
- The Maths Song The GCSE maths topics, in order.
Where to read more
Last reviewed 2026-08-28 by Fiducia Together · Next review due 2027-08-28
Please note: This page explains a word or a process in education. It is general information about the system, the law and the guidance in England — not advice about your child, your school or your exam. If a decision is going against you, get advice from one of the free services we link to before you agree to anything or miss a deadline. If you are revising, check your own exam board's specification: boards word things differently and yours is the one that marks you.