Maths Proof - Prove That If You Add An Odd And Even Number Together, You Get Another Odd Number

In this video you will prove that the sum of an even number and odd number is always an odd number. To do this call the even number 2n and the odd number 2n-1. Since sum means add then 2n+2n-1 which is 4n-1. Now 4n is a multiple of 2 since 4n=2(2n) and so an even number. So if you subtract 1 from it then it will be odd, hence you’ve proved that the sum of an even and odd number is odd.

[carolinehargettsmith]

The way you are proving this is that you're assuming that they are consecutive integers. It would be more accurate if you said the even integer is 2n and the odd integer is 2k + 1 for some integers n and k. If you add these, you would be 2n + 2k + 1 = 2(n + k) + 1. Hence, odd.