Thread: Prove x^2 + x is even?

Prove that for every integer x, is even.

This is what i have so far:

Prove by contradiction,

Suppose is ODD

(where k is any integer)

now plug in (2k + 1) for is ODD

= ODD

= ODD

= ODD

This is False!, ex. when k = 1, is EVEN

Therefore is EVEN

Does this all seem correct? i am very new to proofs. any criticism is appreciated

Thank you

Originally Posted by mybrohshi5

Prove that for every integer x, is even.

This is what i have so far:

Prove by contradiction,

Suppose is ODD

(where k is any integer)

now plug in (2k + 1) for is ODD

= ODD

= ODD

= ODD

This is False!, ex. when k = 1, is EVEN

Therefore is EVEN

Does this all seem correct? i am very new to proofs. any criticism is appreciated

Thank you

I don't think what you have follows. If I read it right you assume by way of contradiction that

is odd this implies that
for some

This does not imply that .

Why not look at it this way



Now just look at two cases
what x is even
and what if x is odd

either way notice that no matter what is either or is even and so is there product.

Here is another way.
For any two consecutive integers, one is even the other is odd.
Even times an odd is even.