Proof (Intuition) Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

Build intuition for the statement: 'For any integer

n

n(n+1)

is even.' Explain informally why this must be true.

1
Intuition: among any two consecutive integers $n$ and $n+1$ , one must be even and one odd.
2
An even number times any integer is even. So $n(n+1)$ always contains at least one even factor.
3
Formal check: if $n$ is even, $n = 2k$ , so $n(n+1) = 2k(n+1)$ is even. If $n$ is odd, $n+1$ is even, so again the product is even.

Answer

n(n+1) \text{ is always even — one of any two consecutive integers is even}

The intuition ('consecutive integers include one even') both explains the result and points directly to the case-split needed in the formal proof.

About Proof (Intuition)

The informal, intuitive sense of why a mathematical statement must be true — the "aha" that precedes and motivates a formal proof.

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