Question

Prove that n^2 +(n+1)^2 is always an odd number

Answer 1

I assume
`n\in\mathbb{Z}` (or at least
`n\in\mathbb{N}`).

`n^2+(n+1)^2=n^2+n^2+2n+1=2n^2+2n+1=2(n^2+n)+1`

The above is always odd, because one of the factors of
`2(n^2+n)` is even, which makes the product even as well, and 1 is odd, and the sum of an even number and an odd number is an odd number.