I'm really not clear why you're trying to do here.
By induction hypothesis, we "know stuff" about
u2m and
u2m+1. We want to prove stuff about
u2m+2 and
u2m+3.
Since the given recurrence doesn't let use find
un+2 without knowing
un,un+1, it's pretty clear that we can only go one step at a time. That is, any attempt to jump straight to
u2m+3 is likely to fail.
Instead, we use the given recurrence to write
u2m+2 in terms of
u2m+1 and
u2m. We then use our induction hypothesis to show
u2m+2=(b/a)u2m+1.
Then we use the given recurrence to write
u2m+3 in terms of
u2m+2 and
u2m+1. And then use our IH and what we've just proved to show that
u2m+3=cu2m+2