So I came across a proof:
(i) $2^n$ is the sum of two consecutive integers.
Proof:
The relation $(2k - 1) + (2k + 1) = 2^n$ implies $k = 2^{n-2}$ and we obtain $2^n = (2^{n-1} - 1) + (2^{n-1} + 1)$.
My question:
Why is the assumption $(2k - 1) + (2k + 1) = 2^n$ to deduce that $k = 2^{n-2}$ valid? Aren't we trying to prove that $(2k - 1) + (2k + 1) = 2^n$?
The proof as presented has several problems. Personally, I'd prefer something like this:
Claim. If $n\ge 2$, then $2^n$ is the sum of two consecutive odd integers.
Proof. As $n\ge 2$, it follows that $k=2^{n-2}$ is an integer, and then $2k-1$ and $2k+1$ are consecutive odd integers. As $(2k-1)+(2k+1)=4k=2^n, $ the claim follows. $\square$
The formulation "implies $k=2^{n-2}$" goes the wrong way and belongs rather to a research attempt to find a proof.