Writing proofs by induction with divisibility is a bit trickier than with sequences, series, and matrices because:
1. you don’t have a formula directly connecting the (k+1)’th thing to the k’th
2. what you do in Step 2 is definitely a chain of reasoning – a series of statements each one of which follows logically from the ones before – and not a chain of working – a series of expressions each of which is calculated to be equal to the one before. And you’re more used to chains of working, though in fact chains of reasoning are much more important in real-life maths.
Anyway, if it’s not clear in your mind, this may help: three methods compared for the example worked through in the textbook on page 127.