Algebraic proof is about using algebra — not just examples — to show that a statement is always true. At GCSE you prove things like "the sum of two consecutive integers is always odd" or "this expression is always a multiple of 6," by building and simplifying expressions with letters.
Why can't you prove something just by trying examples?
Testing a statement for several values shows it might be true, but never proves it is always true. No matter how many examples you check, you cannot rule out a counterexample you haven't tried.
Algebra solves this problem: a correctly constructed algebraic argument covers every possible value of the variable at once. This is the power of proof.
However, to disprove a statement, you only need one counterexample. If a question asks you to "prove or disprove," finding a single value where the statement fails is a complete, valid argument.
What are the key expressions for odd, even, and consecutive numbers?
Memorising these building blocks saves time in the exam.
| What you need | Algebraic expression | Notes |
|---|---|---|
| Any even number | 2n | n is any integer |
| Any odd number | 2n + 1 (or 2n − 1) | One more than an even number |
| Two consecutive integers | n and n + 1 | Or n − 1 and n |
| Three consecutive integers | n, n + 1, n + 2 | — |
| Two consecutive even numbers | 2n and 2n + 2 | Both are multiples of 2 |
| Two consecutive odd numbers | 2n + 1 and 2n + 3 | Gap of 2 |
| A multiple of k | kn | n is any integer |
The choice of which variable (n, m, p …) doesn't matter, but if a proof uses two separate integers, use different letters — using n for both implies they are the same number.
How do you prove the sum or product of integers?
Worked example 1: Prove that the sum of any two consecutive odd numbers is divisible by 4.
Let the two consecutive odd numbers be 2n + 1 and 2n + 3.
Their sum = (2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1)
Since 4(n + 1) is a multiple of 4 for any integer n, the sum of any two consecutive odd numbers is always divisible by 4. □
Worked example 2: Prove that the product of two consecutive integers is always even.
Let the two consecutive integers be n and n + 1.
Product = n(n + 1)
Either n is even (so n = 2k, and the product = 2k(n+1), a multiple of 2), or n is odd (so n + 1 is even, and the product = n × 2k, again a multiple of 2). In both cases the product is even. □
Alternatively: n(n + 1) = n² + n. Notice that this equals 2 × (n² + n)/2 only if n² + n is always even — which follows because one of n or n+1 must be even. Either approach is valid.
How do you prove an expression is always a multiple of a given number?
You need to factorise your final expression so the given number is a visible factor.
Worked example: Prove that n² + 3n + 2 is always even for any positive integer n.
- Factorise: n² + 3n + 2 = (n + 1)(n + 2).
- These are two consecutive integers. As shown above, the product of two consecutive integers is always even.
- Therefore n² + 3n + 2 is always even. □
You could also argue: either n is even (so n + 2 is even, product is even) or n is odd (so n + 1 is even, product is even).
How do you prove that one expression is always greater or less than another?
These "always greater than" proofs often use the fact that a square is always non-negative (x² ≥ 0 for all real x).
Worked example: Prove that (n + 1)² > n² for all positive integers n.
(n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1
For any positive integer n, 2n + 1 > 0, so (n + 1)² > n². □
Another common technique: complete the square to show an expression is always positive or always greater than a given value.
What does a clear proof look like in GCSE mark scheme terms?
GCSE markers look for three things:
- Correct set-up: appropriate algebraic representations of the numbers (e.g. 2n + 1 for odd).
- Correct algebra: expanding, simplifying, and factorising without errors.
- A concluding statement: "Therefore the expression is always a multiple of 3 for any integer n."
The concluding sentence is essential — without it, you have shown it for one arrangement but not explicitly stated the general conclusion. Always finish with a sentence that connects your algebra back to the original claim.
Frequently asked questions
Can I use n = 1, 2, 3 to check my algebraic answer?
Yes — substituting small values is an excellent way to check your algebra before writing up the proof. If your simplified expression gives an odd number for n = 1 when you expect even, there is a slip somewhere. But checking examples is never sufficient as a proof itself.
What does □ mean at the end of a proof?
The small filled or empty square (□ or ■) is a traditional symbol meaning "end of proof" or QED ("quod erat demonstrandum," Latin for "which was to be demonstrated"). You are not required to use it at GCSE, but it is neat and signals clearly to the marker that your argument is complete.
How do I prove something is NOT always true?
Provide a counterexample — one specific value for which the statement fails. For example, to disprove "n² + n + 41 is always prime," try n = 41: 41² + 41 + 41 = 41(41 + 1 + 1) = 41 × 43, which is not prime. One counterexample is enough; you do not need to explain why it fails in general.
Does algebraic proof appear on Foundation tier?
Basic versions — such as "show this expression is even" or "disprove by counterexample" — can appear on Foundation Higher. Full proof involving factorising and showing divisibility is mostly a Higher tier topic. Check your specification, but the key expressions for odd, even, and consecutive numbers are useful at both tiers.
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