Algebraic Proof GCSE – Step-by-Step Guide

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 five marks. This guide covers the key representations, a reliable method, and fully worked examples so you can approach proof questions with confidence.

What Is Algebraic Proof?

An algebraic proof uses algebra to demonstrate that a statement is always true for all values. Unlike a numerical check (which only shows it works for one number), a proof shows it must work in every case.

Key representations

To write proofs, you need to express types of numbers algebraically:

Type of number	Algebraic form
Any integer	n
Consecutive integers	n, n + 1, n + 2, ...
Even number	2n
Odd number	2n + 1
Consecutive even numbers	2n, 2n + 2
Consecutive odd numbers	2n + 1, 2n + 3
Multiple of 3	3n
Square number	n²

Important: If a proof involves two independent integers, use different letters (e.g., n and m), not the same letter.

What the exam wants to see

A clear algebraic expression for each number.
Manipulation (expanding, simplifying, factorising).
A concluding statement explaining why the result proves the claim.

Step-by-Step Method

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 1 — Foundation-style Proof

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:

n + (n + 1) + (n + 2) = 3n + 3

Step 3: Factorise:

3n + 3 = 3(n + 1)

Step 4: Conclusion:

3(n + 1) is a multiple of 3 for all integer values of n. Therefore the sum of three consecutive integers is always a multiple of 3. ■

Worked Example 2 — Higher Level

Question: Prove that the difference between the squares of two consecutive odd numbers is always a multiple of 8.

Step 1: Let the two consecutive odd numbers be (2n + 1) and (2n + 3).

Step 2: Find the difference of their squares:

(2n + 3)² − (2n + 1)²

Step 3: Expand each bracket:

(2n + 3)² = 4n² + 12n + 9

(2n + 1)² = 4n² + 4n + 1

Step 4: Subtract:

(4n² + 12n + 9) − (4n² + 4n + 1) = 8n + 8

Step 5: Factorise:

8n + 8 = 8(n + 1)

Step 6: Conclusion:

8(n + 1) is a multiple of 8 for all integer values of n. Therefore the difference between the squares of two consecutive odd numbers is always a multiple of 8. ■

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.

Practice Questions

Question 1: Prove that the sum of two consecutive even numbers is always even but not a multiple of 4.

Answer: Let the two consecutive even numbers be 2n and 2n + 2. Sum = 2n + 2n + 2 = 4n + 2 = 2(2n + 1). This is 2 times an odd number, so it is even. Since 2n + 1 is odd, it cannot be written as 2k, so 2(2n + 1) is not a multiple of 4.

Question 2: Prove that the product of two odd numbers is always odd.

Answer: Let the two odd numbers be (2n + 1) and (2m + 1). Product = (2n + 1)(2m + 1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1. This is in the form 2k + 1, which is odd.

Question 3: Prove that n² + n is always even for any integer n.

Answer: n² + n = n(n + 1). This is the product of two consecutive integers. One of them must be even, so the product is always even.

Put your proof skills to the test with unlimited practice on GCSEMathsAI. Our AI marks your working step by step and flags exactly where you might lose marks.

Summary

Algebraic proof shows a statement is true for all values, not just specific examples.
Represent even numbers as 2n, odd numbers as 2n + 1, and consecutive integers as n, n + 1, n + 2.
Use different letters (n and m) for independent values.
Follow a clear structure: represent, manipulate, factorise, conclude.
Always write a concluding sentence that explicitly states why the algebra proves the claim.
Common proofs involve sums, differences and products of odd/even/consecutive numbers.
The difference of two squares identity a² − b² = (a + b)(a − b) is frequently useful.

Algebraic Proof