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Algebraic proof
A proof shows something is true for every number, not just the examples you tried.
Useful forms
Any integer: n. Even number: 2n. Odd number: 2n + 1.
Consecutive integers: n, n + 1, n + 2.
A multiple of 5: 5 × (something).
Consecutive integers: n, n + 1, n + 2.
A multiple of 5: 5 × (something).
Structure of a proof
1. Write the numbers algebraically.
2. Expand and simplify.
3. Factorise to show the required form.
4. Write a conclusion.
2. Expand and simplify.
3. Factorise to show the required form.
4. Write a conclusion.
Example
Prove the sum of two consecutive odd numbers is a multiple of 4.
(2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1)
4(n + 1) is a multiple of 4.
(2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1)
4(n + 1) is a multiple of 4.
Prove that (n + 3)2 − (n − 3)2 is a multiple of 12.
- Expand: n2 + 6n + 9 − (n2 − 6n + 9)
- = 12n
- 12n = 12 × n, a multiple of 12
Answer: Shown: 12n
Write numbers algebraically (2n even, 2n + 1 odd), expand and simplify, factorise to show the form, and state the conclusion.
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