This approach has been explored. It has only been helpful for showing that certain types of (short-ish) cycles can't occur [1]. Showing that the sequence can't

This approach has been explored. It has only been helpful for showing that certain types of (short-ish) cycles can't occur [1]. Showing that the sequence can't get larger and larger forever (without repeating) with this approach has been a non-starter. [1] See https://en.wikipedia.org/wiki/Collatz_conjecture#Cycle_length https://en.wikipedia.org/wiki/Collatz_conjecture#Cycle_lengt... for a few references. The most recent appears to be the paper "Theoretical and computational bounds for m-cycles of the 3n + 1 problem" by Simons and de Weger.