Proofs Math Example 2

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

Example 2

medium
Prove directly: For any integer nn, if nn is odd then n2n^2 is odd.

Solution

  1. 1
    Assume nn is odd, so n=2k+1n = 2k + 1 for some integer kk.
  2. 2
    Compute n2=(2k+1)2=4k2+4k+1=2(2k2+2k)+1n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1.
  3. 3
    Let m=2k2+2km = 2k^2 + 2k, which is an integer. Then n2=2m+1n^2 = 2m + 1, which is odd.

Answer

n2=2(2k2+2k)+1 is odd.n^2 = 2(2k^2+2k)+1 \text{ is odd.}
The proof uses the definition of odd numbers (2k+12k+1 form) and algebra to show the square retains that form. This is a classic example of direct proof structure.

About Proofs

A mathematical proof is a rigorous logical argument that demonstrates the truth of a statement beyond doubt, proceeding from accepted axioms and previously proven results through valid inference rules.

Learn more about Proofs →

More Proofs Examples

Example 1 easy

Prove directly: The sum of two even integers is even.

Example 3 medium

Prove: The product of two odd integers is odd.

Example 4 medium

Prove directly: if [formula] is even, then [formula] is odd.

← All Proofs Examples Proofs Concept