In algebraic proof, write an even number as 2k and an odd number as 2n + 1, where n and k are integers. Free notes and 10 questions answered.
Updated 21 Mar 2026 · Revise Right Team
Algebraic Proofs is part of Notation, vocabulary and manipulation in GCSE AQA Higher Maths.
4 min
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GCSE AQA Higher
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What this lesson covers
In algebraic proof, write an even number as 2k and an odd number as 2n + 1, where n and k are integers. Free notes and 10 questions answered.
Define what constitutes a valid algebraic proof and explain why general algebraic arguments are required
Use algebraic expressions to represent general integers, consecutive integers, odd and even numbers
Construct and present a complete algebraic proof for statements about number properties
Identify and correct common errors in attempted algebraic proofs
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Use this lesson to stay inside Notation, vocabulary and manipulation while connecting the detail back to the wider Algebra topic.
Algebraic proof: A logical argument using algebra to show that a statement is true for all values, not just specific examples. Used when a result must be demonstrated generally rather than by testing cases.
Consecutive integers: Integers that follow one another in order with no gaps. Represented algebraically as $n$, $n+1$, $n+2$,... where $n$ is any integer.
Even and odd integers: An even integer is any integer of the form $2n$ where $n$ is an integer. An odd integer is any integer of the form $2n+1$ where $n$ is an integer.
Identity: An equation that is true for all values of the variable(s), written using the $\equiv$ symbol. Distinct from an equation, which is only true for specific values.
Revision notes
Read the core explanations from Algebraic Proofs before testing yourself from memory.
1. Why Proof Matters — and What Makes It Different
Here is a claim: "The sum of any two odd numbers is always even." You could check it with examples — $3 + 5 = 8$, $7 + 11 = 18$, $1 + 99 = 100$ — and every one works. But does that prove it? No. Testing examples only shows it works for those cases. A proof must show it works for every possible pair of odd numbers, including ones nobody has ever tried.
That is the heart of algebraic proof: replacing specific numbers with general expressions so the argument covers all cases at once. When Exam questions ask you to "prove" or "show that", they want a chain of algebraic steps that leaves no room for doubt.
2. The Language of Proof — Representing Numbers Algebraically
Before you can prove anything, you need to express numbers in general form. This is the toolkit every proof question expects you to reach for immediately.
Notice that two different odd numbers should use the same letter $n$ only if they are related (like consecutive odds). If the question says "any two odd numbers", use $2m+1$ and $2n+1$ with different letters, so they are genuinely independent.
3. Constructing a Proof — The Method Step by Step
Every algebraic proof follows the same structure: start with general expressions, manipulate them algebraically, and show the result matches the required form. The key discipline is that you must start from the given information and work towards the conclusion — never work backwards from the answer.
Step 1 — Define: Write general algebraic expressions for the numbers described.
Step 2 — Form: Write the expression for the quantity you are proving something about (e.g. the sum, product, or difference).
Step 3 — Expand and simplify: Use algebra to rewrite the expression in a revealing form.
Step 4 — Conclude: Show the result matches the required form and state your conclusion clearly.
The conclusion sentence is worth a mark in its own right on GCSE Higher exam papers. Always write a final statement such as "This is of the form $2k$, where $k$ is an integer, so the result is always even."
Worked example: Proof: The Sum of Two Odd Numbers is Always Even
Prove that the sum of any two odd numbers is always even.
Let the two odd numbers be $2m + 1$ and $2n + 1$, where $m$ and $n$ are integers. — Using different letters $m$ and $n$ ensures the two odd numbers are completely independent — they do not have to be consecutive or equal. [M1 for correct general form]
Sum $= (2m + 1) + (2n + 1)$ — Write the expression for what we are proving — the sum of the two numbers.
$= 2m + 2n + 2$ — Collect the integer terms: $1 + 1 = 2$. [M1 for correct expansion]
$= 2(m + n + 1)$ — Factor out 2. Since $m$ and $n$ are integers, $m + n + 1$ is also an integer. [A1 for correct factorisation]
This is of the form $2k$ where $k = m + n + 1$ is an integer. — Any number of the form $2k$ is even by definition. The conclusion sentence secures the final mark. [A1 for conclusion]
Answer: The sum of any two odd numbers is always even. (QED)
Worked example: Proof: The Product of Two Consecutive Integers is Always Even
Prove that the product of two consecutive integers is always even.
Let the two consecutive integers be $n$ and $n + 1$, where $n$ is an integer. — Consecutive integers differ by 1. Starting with $n$ covers all possible pairs. [M1]
Product $= n(n + 1)$ — Write the expression for the product of the two consecutive integers.
Case 1: If $n$ is even, then $n = 2k$, so $n(n+1) = 2k(n+1)$, which is a multiple of 2.
Case 2: If $n$ is odd, then $n + 1$ is even, so $n + 1 = 2k$, giving $n(n+1) = n \cdot 2k$, which is a multiple of 2. — In any pair of consecutive integers, exactly one must be even. Multiplying by an even number always gives an even result. [M1 for valid reasoning]
In both cases, $n(n+1)$ is a multiple of 2, so the product of two consecutive integers is always even. — A clear conclusion referencing both cases secures full marks. [A1]
Answer: The product of any two consecutive integers is always even. (QED)
4. Proving Algebraic Identities
Proof questions are not limited to number properties. Exam questions also ask you to prove algebraic identities — statements that two expressions are always equal. Here the approach is to manipulate one side (usually the more complex side) until it matches the other, using expansion and factorisation.
Worked example: Proving an Algebraic Identity
Show that $(n + 3)^2 - (n + 1)^2 \equiv 4(n + 2)$.
$(n^2 + 6n + 9) - (n^2 + 2n + 1)$
$= n^2 + 6n + 9 - n^2 - 2n - 1$
$= 4n + 8$ — Take care with the minus sign — it applies to every term in the second bracket. [M1 for correct subtraction]
$4n + 8 = 4(n + 2)$ — Factor out 4. This matches the right-hand side exactly. [A1]
$(n + 3)^2 - (n + 1)^2 = 4(n + 2)$ for all values of $n$, as required. — Confirm both sides are equal and that the identity holds generally. This conclusion sentence secures the final mark.
The only way to get confident with proof is to work through questions yourself. The problems below follow the same structure as GCSE Higher exam questions. Try each one before checking the solution.
6. Exam Technique — Marks, Mistakes, and What Examiners Look For
Proof questions on GCSE Higher exam papers are typically worth 3–4 marks, and the marks are awarded for method, algebraic accuracy, and a clear conclusion. Losing the conclusion mark is the single most common way students drop a mark they have otherwise earned. Remember that algebraic proof consolidates key algebra skills and proves essential for understanding mathematical arguments.
mark schemes also reward correct general form at the start. If you write $n$ for an odd number instead of $2n+1$, you cannot recover the proof — the algebra will not lead anywhere useful. Getting the definitions right at Step 1 protects all the marks that follow.
Algebraic proof brings together the core algebra skills from across the GCSE course — expanding brackets, factorising, and working with general expressions — and applies them to construct watertight mathematical arguments. Master the four-step structure (define, form, simplify, conclude) and you have a reliable method for every proof question you may be asked.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-21.
How do you prove the product of two consecutive integers is even?
Let the consecutive integers be n and n + 1, so the product is n(n + 1). In any pair of consecutive integers exactly one must be even, so in both cases the product is a multiple of 2 and is always even.
How do you write an odd number in algebra?
An odd number is written as 2n + 1, where n is an integer, and any number of the form 2k is even. For any two odd numbers that are not related, use different letters such as 2m + 1 and 2n + 1 so they are genuinely independent.
What are the steps of an algebraic proof?
An algebraic proof has four steps: define general expressions for the numbers, form the expression for the sum, product or difference, expand and simplify it into a revealing form, then conclude with a clear statement, such as this is of the form 2k, so it is always even.
How do you write an even number in algebra?
An even number is written in algebra as 2k, where k is an integer, because any number of the form 2k is even by definition. A proof that a result is even ends by showing it can be written in this form, such as 2(m + n + 1).
How do you write consecutive integers in algebra?
Consecutive integers are written in algebra as n and n + 1, where n is an integer, because consecutive integers differ by 1. Starting with n covers all possible pairs, so a proof using n(n + 1) works for every pair.
Why use different letters for two odd numbers in a proof?
Different letters such as 2m + 1 and 2n + 1 are used for any two odd numbers so that they are genuinely independent. Using the same letter n would only cover related numbers, like consecutive odds, so the proof would not cover every case.
How many marks are algebraic proof questions worth?
Algebraic proof questions are typically worth 3–4 marks, awarded for method, algebraic accuracy and a clear conclusion. Losing the conclusion mark is the single most common way students drop a mark they have otherwise earned.
What are common mistakes in algebraic proofs?
Common mistakes in algebraic proofs are using only examples, writing n for an odd number instead of 2n + 1, working backwards from the answer and leaving out the conclusion. Getting the general form right at the start protects all the marks that follow.
Why do you need a conclusion in an algebraic proof?
A conclusion is needed in an algebraic proof because the conclusion sentence is worth a mark in its own right. A final statement such as 'This is of the form 2k, where k is an integer, so the result is always even' shows the result matches the required form.
How do you prove an algebraic identity?
An algebraic identity is proved by manipulating one side, usually the more complex side, with expansion and factorisation until it matches the other. For example, (n + 3)² - (n + 1)² = n² + 6n + 9 - n² - 2n - 1 = 4n + 8 = 4(n + 2).