[Here's](https://math.stackexchange.com/questions/4213102/how-to-arrive-at-the-nth-term-of-the-following-progression) the related question

Is it possible to prove there exists at least one positive integer solution for $k$ and $n$ in $2^{k} = 3^{n-1}(m + 1) - 1$, where m is any even positive integer?

So we need to prove, that for any given even positive integer m, there exists at least one positive integer solution of k and n.

Eg:  

If m = 2, then k = 3, n = 2  

If m = 4, then k = 2, n = 1