Base representation problem

Question

This is a very simple problem, but the solution for it does not make too much sense to me, what I believe should be the solution is that since we want to represent it in terms of (b-1) we should have it so it is (b-1)(b-2) + 2*(b-1)^0 yet for some reason the solution is (b-1)(b-2) + (b-2) + 2.

Answer 1

Your number is $(b-2)b+2$, which is equal to $(b-1)^2+1$. In other words, it is equal to $1\times(b-1)^2+0\times(b-1)+1$. Therefore, its representation in the base $b-1$ is $101$.