Direct Proof Math Example 4

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

Example 4

medium
Prove directly: for all integers nn, n2+nn^2 + n is even.

Solution

  1. 1
    Note that n2+n=n(n+1)n^2 + n = n(n+1) — the product of two consecutive integers.
  2. 2
    Among any two consecutive integers, one is even. If nn is even, n=2kn = 2k, so n(n+1)=2k(n+1)n(n+1) = 2k(n+1) is even.
  3. 3
    If nn is odd, n+1n+1 is even, n+1=2kn+1 = 2k, so n(n+1)=n(2k)n(n+1) = n(2k) is even.
  4. 4
    In both cases n(n+1)n(n+1) is even. □\square

Answer

n2+n=n(n+1) is always even□n^2+n = n(n+1) \text{ is always even} \quad \square
Factoring n2+n=n(n+1)n^2+n = n(n+1) reveals the consecutive-integer structure, making the proof immediate. The case split (n even / n odd) is a standard direct-proof technique for statements about integers.

About Direct Proof

A direct proof establishes a statement P⇒QP \Rightarrow Q by assuming PP is true and using logical steps, definitions, and known theorems to arrive at QQ — the most straightforward proof strategy.

Learn more about Direct Proof →

More Direct Proof Examples

Example 1 easy

Prove directly: the sum of an even integer and an odd integer is odd.

Example 2 medium

Prove directly: for all real [formula], [formula].

Example 3 easy

Prove directly: if [formula], then [formula].

← All Direct Proof Examples Direct Proof Concept