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

The Answer to the Question
is below this banner.

Can't find a solution anywhere?

NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

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.