I have a lot of difficulty understanding how these proofs were figured out. I know that most of the proofs in the world just explain how they get from point A to Z but don't explain how they thought of going through all the other 24 points.
In class today, we learned that \for all natural numbers, n² + n is even.
It was done using proof by cases where n² + n had to be even regardless of if n was odd of if n was even.
However, one person pointed out what many were probably thinking in class:
if n² + n is factorized into n(n+1), this intuitively explains that it must be even because either n or n+1 must be even and any natural number multiplied by an even number is always even.
The problem is that I don't know how to prove this without using the proof by cases and substituting n with 2k+1 or 2k.
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