Answer to Question #142684 in Discrete Mathematics for Christine Joy Sato

Question

Statement: “If n is any even integer, then ”
Using Direct proof method

Answer 1

Prove the statement: if n is any even integer, then (−1)ⁿ = 1.

Proof:

Suppose n is even integer. Then by the definition of even numbers, n = 2k for some integer k.

Hence, by the laws of exponents of algebra, we have

$$(-1)^n=(-1)^{2k}=((-1)^2)^k=(1)^k=1$$

This is what was to be shown.

And this completes the proof.