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Direct proof

A direct proof starts from what you know and works its way to the claim. Three consecutive integers can be written $n$, $n + 1$ and $n + 2$. The sum becomes $3(n + 1)$, which is always divisible by 3. The same setup solves a number problem: if the sum is known, the middle number is the sum divided by 3. The proof holds for every integer, not only for one example.

Symbols

Example

The sum $13 + 14 + 15 = 42$. In general the middle number is $\dfrac{42}{3} = 14$, and the sum $3(n + 1)$ is divisible by $3$.

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Part of Mathematics R1: Logic and proof.