Show that the sum of any two consecutive integers can be written as either $4n + 1$ or $4n − 1$ for some integer $n$.
I know this is a proof by cases. I'm having trouble proving the $4n-1$ part.
2026-03-25 14:23:44.1774448624
Proof of sum of consecutive integers
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Any two consecutive integers added together will be odd. This is because any two consecutive integers have opposite parity. One number is even, and the other is odd. An even number plus an odd number will always be odd.
$4n+1$ and $4n-1$ are both odd.
This is not a formal proof, but this should definitely be enough to get you started!