The simplest method of proof is
proof by deduction, where you start from known facts or definitions, then showing by deduction that another statement is true (or untrue).
Statements that can be assumed to be true include "all even numbers can be written as \(2n\)" and "all odd numbers can be written as \(2n+1\)".
Proof by deduction involves:
- Start with a statement, and assume that it is true.
- Use this statement to show that another statement must be true.
- Finish with a statement of proof.
Prove that the sum of the squares of two consecutive integers is odd.
Let the first integer be \(n\), and the second integer be \(n+1\).
\[n^2 + (n+1)^2\]\[= n^2 + n^2 + 2n + 1\]\[= 2n^2 + 2n + 1\]\[= 2(n^2 + n) + 1\]
Since \(2(n^2 + n)\) must be even, and an even number \(+1\) is odd, so therefore the sum of the squares of two consecutive integers is odd.
Prove that \(n^2 - n\) is an even number for all values of \(n\).
\[n^2 - n = n(n - 1)\]
\(n\) can only be even or odd.
If \(n\) is even, then \(n-1\) must be odd. Even \(×\) odd \(=\) even.
If \(n\) is odd, then \(n-1\) must be even. Odd \(×\) even \(=\) even.
Therefore \(n^2 - n\) is an even number for all values of \(n\).
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