In proof, checking a few examples is not enough. We use a letter such as $n$ to stand for any allowed integer, so the argument works for every case.

Standard forms matter. Even numbers are $2n$, odd numbers are $2n+1$, and consecutive integers can be written as $n$ and $n+1$.

Before you start, decide the target form you want to reach, such as $2k+1$, $6k$, $k^2$, or $8k+2$. A good proof follows a clear chain: define the general number, substitute, simplify, factorise if needed, and conclude. The last sentence must explain why the remaining part is an integer.