A6: Algebraic Proof
Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent
📋 Key Concepts
Equation vs Identity:
- Equation: True for specific values (e.g., 2x = 6 is true only when x = 3)
- Identity: True for ALL values, shown with ≡ symbol (e.g., 2(x + 3) ≡ 2x + 6)
Algebraic Proof: Using algebra to show that a mathematical statement is always true.
📝 Proving Identities
Method: Expand and simplify one or both sides to show they are identical.
Example 1
Prove that (x + 3)² + (x - 3)² ≡ 2(x² + 9)
Solution:
LHS = (x + 3)² + (x - 3)²
= (x² + 6x + 9) + (x² - 6x + 9)
= x² + 6x + 9 + x² - 6x + 9
= 2x² + 18
= 2(x² + 9)
= RHS ∎
Example 2
Prove that (n + 5)² - (n - 5)² ≡ 20n
Solution:
LHS = (n + 5)² - (n - 5)²
= (n² + 10n + 25) - (n² - 10n + 25)
= n² + 10n + 25 - n² + 10n - 25
= 20n
= RHS ∎
📝 Algebraic Expressions for Numbers
Useful expressions:
| Phrase | Expression |
|---|---|
| Any integer n | n |
| Even number | 2n |
| Odd number | 2n + 1 or 2n - 1 |
| Consecutive integers | n, n + 1, n + 2... |
| Consecutive even numbers | 2n, 2n + 2, 2n + 4... |
| Consecutive odd numbers | 2n + 1, 2n + 3, 2n + 5... |
| Multiple of 3 | 3n |
| Square of n | n² |
Example 3
Prove that the sum of two consecutive even numbers is always a multiple of 4.
Solution:
Let the consecutive even numbers be 2n and 2n + 2
Sum = 2n + (2n + 2)
= 4n + 2
= 2(2n + 1)
This is NOT a multiple of 4 - the conjecture is false!
Counter-example: 4 + 6 = 10, which is not divisible by 4
Example 4
Prove that the sum of three consecutive integers is always a multiple of 3.
Solution:
Let the integers be n, n + 1, n + 2
Sum = n + (n + 1) + (n + 2)
= 3n + 3
= 3(n + 1)
Since 3(n + 1) is divisible by 3 for any integer n, the statement is proven ∎
📝 Proving with Squares
Example 5
Prove that the difference between the squares of two consecutive integers is always odd.
Solution:
Let the consecutive integers be n and n + 1
Difference = (n + 1)² - n²
= (n² + 2n + 1) - n²
= 2n + 1
Since 2n is even, 2n + 1 is always odd ∎
📝 Disproving Conjectures
Counter-example: To disprove a statement, find ONE example where it doesn't work.
Example 6
Prove or disprove: "All square numbers are even."
Solution:
Counter-example: 9 = 3², but 9 is odd.
The statement is false.
Example 7
Prove or disprove: "n² + n + 41 is always prime for any integer n."
Solution:
Try n = 41:
41² + 41 + 41 = 41(41 + 1 + 1) = 41 × 43
This is NOT prime. The conjecture is false.
❓ Practice Questions
Q1: Prove that (x + 4)² - (x + 2)² ≡ 4(2x + 3)
Q2: Write an expression for two consecutive odd numbers and find their sum.
Q3: Prove that the sum of three consecutive even numbers is divisible by 6.
Q4: Prove that (2n + 1)² - (2n - 1)² = 8n
Q5: Find a counter-example to disprove: "All prime numbers are odd."
Q6: Prove that the product of two consecutive integers is always even.
✅ Answers
- LHS = x² + 8x + 16 - (x² + 4x + 4) = 4x + 12 = 4(2x + 3) = RHS ✓
- 2n + 1 and 2n + 3, Sum = 4n + 4
- Let numbers be 2n, 2n + 2, 2n + 4. Sum = 6n + 6 = 6(n + 1) ✓
- LHS = (4n² + 4n + 1) - (4n² - 4n + 1) = 8n = RHS ✓
- 2 is prime and 2 is even
- Let numbers be n and n + 1. One must be even (if n is odd, n + 1 is even; if n is even, n is even). Product of any even number is even ✓
🎯 Exam Tips
- Use ≡ for identities, = for equations
- Define your variables clearly (Let n be...)
- Show every step of your working
- End proofs with QED or ∎
- To disprove, find ONE counter-example
- Know standard expressions: 2n (even), 2n+1 (odd)
🧠 Problem-Solving Strategies
Problem-Solving
Start proofs by defining variables with clear algebraic expressions (e.g., 2n for even, 2n+1 for odd). Show every step of working. For identities, expand both sides. For disproof, find one counter-example. Always end with QED or ∎.
Multi-Step Problem
Prove that the product of two odd numbers is always odd.
Solution:
Let the two odd numbers be 2m + 1 and 2n + 1
Product = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1
Since 2(2mn + m + n) is even (it's 2 × something), adding 1 makes it odd. ∎
⚠️ Common Errors
Watch Out!
1. Wrong: Using n and n + 1 for consecutive odd numbers Correct: Consecutive odd numbers are 2n + 1 and 2n + 3
2. Wrong: Proving an identity by substituting one value Correct: Must show both sides are algebraically identical, or test multiple values
3. Wrong: Saying "it works for n = 1, 2, 3 so it's proven" Correct: This only shows a pattern — a proof must work for ALL values using algebra
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Prove that the difference between the squares of any two consecutive even numbers is always a multiple of 4.
Let the consecutive even numbers be 2n and 2n + 2.
Their squares are (2n)² = 4n² and (2n + 2)² = 4n² + 8n + 4
Difference = (2n + 2)² - (2n)² = 4n² + 8n + 4 - 4n² = 8n + 4 = 4(2n + 1)
Since 4(2n + 1) has a factor of 4 for any integer n, the difference is always a multiple of 4. ∎
Mark scheme: 1 mark for correct expressions, 1 mark for squaring, 2 marks for correct subtraction, 1 mark for factorising, 1 mark for conclusion.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student claims: "The sum of any three consecutive integers is always even."
(a) Write an expression for the sum of n, n + 1 and n + 2.
(b) Prove or disprove the claim.
(c) What type of number is the sum always a multiple of?
Answers: (a) 3n + 3 = 3(n + 1). (b) Disproved: when n = 1, sum = 6 (even); when n = 2, sum = 9 (odd). The claim is false. (c) The sum is always a multiple of 3, not always even.
📝 Exam Questions by Topic
🎬 Video Resources
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