Question

Show an informal proof to show that the following conjecture is true

Conjecture: The sum of an odd number and an even number is an odd number

Answer 1

I mostly don't really know how to type a informal proof.

Answer 2

Think of the integers.
... , -3, -2, -1, 0, 1, 2, 3, ...

Answer 3

Some integers are even, and some are odd.
If you multiply every integer by 2, are the products odd or even?

Answer 4

even

Answer 5

Correct.

... , 2(-3), 2(-2), 2(-1), 2(0), 2(1), 2(2), 2(3), ...

are all even

Answer 6

Since 2 times each integer is an even integer, we can write

For every integer n, 2n is an even integer.

Answer 7

Then by the same token, we can write

For every integer n, 2n + 1 is an odd integer.

Answer 8

I just really don't understand how to prove that. with an informal proof

Answer 9

We're getting there.
So far do you understand that
for every integer, n, 2n is an even integer, and
for every integer n, 2n + 1 is an odd integer?

Answer 10

Now let's add the numbers.

odd even
What is (2n + 1) + (2n) ?

(2n + 1) + (2n) =
= 2n + 1 + 2n
= 4n + 1
= 2(2n) + 1

Since for every integer n, 2n is an even integer, then 2(2n) must be even.
Therefore, 2(2n) + 1 = 4n + 1 is odd.