Sheet № 31 · Higher only · AQA · Edexcel · OCR
Algebraic Proof
Algebraic proof is one of the most challenging Higher-tier topics at GCSE, yet it follows a clear structure once you know the building blocks. Proof questions ask you to show that a mathematical statement is always true — not just for one example, but for every possible case. They appear on AQA, Edexcel and OCR papers and can carry up to
§Key definitions
Important:
If a proof involves two independent integers, use different letters (e.g., n and m), not the same letter.
Question:
Prove that the sum of any three consecutive integers is always a multiple of 3.
Step 1:
Let the three consecutive integers be n, n + 1 and n + 2.
Step 2:
Find their sum:
Step 3:
Factorise:
§Formulas to memorise
Important:: If a proof involves two independent integers, use different letters (e.g., n and m), not the same letter.
Read the statement carefully. — Identify what type of numbers are involved (even, odd, consecutive, etc.).
Represent — each number algebraically using the table above.
Write the expression — described in the question (e.g., "the sum of three consecutive integers").
Expand and simplify — the algebra.
Factorise or rearrange — to show the required property.
Write a conclusion — that links your algebra to the statement. For example: "This is 2(...), which is even, so the sum is always even."
Worked example
Prove that the sum of any three consecutive integers is always a multiple of 3.
Step 1: Let the three consecutive integers be n, n + 1 and n + 2.
⚠ Common mistakes
- ✗Using examples instead of algebra. Showing that 3 + 5 = 8 is even does not prove it works for all odd numbers. You must use algebraic expressions.
- ✗Using the same letter for independent values. If a question says "any two even numbers", they might not be the same. Use 2n and 2m, not 2n and 2n.
- ✗Forgetting the conclusion. You must write a sentence explaining why your final expression proves the statement. Without this, you will lose the final mark.
- ✗Not factorising fully. If you need to show something is a multiple of 6, you must show 6 as a factor, not just 2 and 3 separately (unless you explicitly explain that 2 × 3 = 6).
- ✗Incorrect expansion. (2n + 1)² = 4n² + 4n + 1, not 4n² + 1. Do not forget the middle term when expanding.
✦ Exam tips
- →Learn the standard representations by heart. Knowing instantly that an even number is 2n and an odd number is 2n + 1 saves time and prevents errors.
- →Use the difference of two squares where appropriate: a² − b² = (a + b)(a − b). This often simplifies proof questions involving squared terms.
- →On AQA papers, the final line must be a clear statement — not just the factorised expression. Write "Therefore ... is always ..." as a full sentence.
- →Practise common proof types: sum/difference of odds and evens, consecutive numbers, and divisibility proofs. These cover the majority of exam questions.