Algebraic proof
General numbers
Checking examples is not a proof: the result must hold for every whole number.
Use letters.
- An even number: 2n. An odd number: 2n + 1.
- Consecutive numbers: n, n + 1, n + 2. Consecutive even numbers: 2n, 2n + 2.
- A multiple of 3: 3n.
Two different odd numbers need two letters: 2a + 1 and 2b + 1.
Expand, simplify, conclude
- prove (n + 3)² − (n − 3)² is a multiple of 12
- (n + 3)² = n² + 6n + 9
- (n − 3)² = n² − 6n + 9
- difference = 12n
- 12n is a multiple of 12, for every n
Finish with a sentence that says why: "12n = 12 × n, so it is a multiple of 12".
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