How can you be sure a mathematical pattern will always hold true? Algebraic proofs use the power of variables to show a statement is true for all possible cases, not just a few examples. It’s a key higher-tier skill for thinking logically and demonstrating mathematical truths.
The Language of Proof
The key to algebraic proofs is being able to represent different types of numbers using variables.
Any Integer
$n$
An Even Number
$2n$
An Odd Number
$2n+1$
Consecutive Integers
$n,\ n+1,\ n+2$
The Structure of a Proof
A good proof follows a clear, logical structure. Think of it in four steps.
1. Define Your Variables
Start by stating what your letters represent. For example, “Let $n$ be any integer” or “Let $a$ and $b$ be integers.”
2. Set Up the Expression
Use your algebraic representations to write down the calculation described in the statement (e.g., the sum, product, or difference).
3. Manipulate the Algebra
Expand, simplify, and factorise your expression until it matches the form you’re trying to prove (e.g., $2 \times (\text{integer})$ for an even number).
4. Write a Conclusion
Finish with a clear sentence that links your algebraic result back to the original statement, explaining why it’s now proven.
Worked Examples
Example 1: Sum of Even Numbers
Prove the sum of two even numbers is always even.
- Let the two even numbers be $2a$ and $2b$, where $a$ and $b$ are integers.
- Their sum is $2a + 2b$.
- Factorise: $2(a+b)$.
- Since $a+b$ is an integer, $2(a+b)$ is a multiple of 2, and therefore is always even. ∎
Example 2: Product of Odd Numbers
Prove the product of two odd numbers is always odd.
- Let the two odd numbers be $2a+1$ and $2b+1$, where $a$ and $b$ are integers.
- Their product is $(2a+1)(2b+1)$.
- Expand: $4ab + 2a + 2b + 1$.
- Factorise: $2(2ab+a+b) + 1$.
- This is in the form $2 \times (\text{integer}) + 1$, which is the definition of an odd number. ∎
Example 3: Consecutive Integers
Prove the difference between the squares of two consecutive integers is always odd.
- Let the integers be $n$ and $n+1$, where $n$ is an integer.
- The difference of their squares is $(n+1)^2 – n^2$.
- Expand: $(n^2 + 2n + 1) – n^2$.
- Simplify: $2n+1$.
- This is in the form $2n+1$, which is the definition of an odd number. ∎
Example 4: Multiples
Prove $(n+2)^2 – n^2$ is always a multiple of 4.
- Let $n$ be any integer. The expression is $(n+2)^2 – n^2$.
- Expand: $(n^2 + 4n + 4) – n^2$.
- Simplify: $4n+4$.
- Factorise: $4(n+1)$.
- Since $n+1$ is an integer, $4(n+1)$ is always a multiple of 4. ∎
Tutor Insights
🤔 Common Misunderstandings
- Examples are not proofs: Showing $3+5=8$ suggests a rule, but doesn’t prove it for all odd numbers. Algebra is needed for that.
- Incorrect algebraic representation: Mixing up $2n$ (even) and $2n+1$ (odd), or using the same letter for two different numbers (e.g., $2n + 2n$ when you mean two different even numbers).
📝 Common Exam Mistakes
- Algebraic errors: Simple mistakes when expanding brackets or simplifying. This is where most marks are lost.
- No concluding statement: You must write a final sentence that links your algebra back to the original statement to complete the proof.
- Forgetting to define variables at the start (e.g., “Let $n$ be an integer”).
Practice Questions
Set out your proofs clearly, defining your variables and writing a conclusion.
- Prove that the sum of two odd numbers is always an even number.
- Prove that the sum of three consecutive integers is always a multiple of 3.
- Prove that $(n+1)^2 – (n-1)^2$ is always a multiple of 4 for any integer $n$.
Show Answers
- Proof: Let the odd numbers be $2a+1$ and $2b+1$, where $a$ and $b$ are integers. Their sum is $(2a+1)+(2b+1) = 2a+2b+2 = 2(a+b+1)$. Since $a+b+1$ is an integer, this is a multiple of 2 and therefore even. ∎
- Proof: Let the integers be $n$, $n+1$, and $n+2$. Their sum is $n+(n+1)+(n+2) = 3n+3 = 3(n+1)$. Since $n+1$ is an integer, this is a multiple of 3. ∎
- Proof: Expand: $(n^2+2n+1)-(n^2-2n+1) = n^2+2n+1-n^2+2n-1 = 4n$. Since $n$ is an integer, $4n$ is always a multiple of 4. ∎
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