Problems with a non-linear recursion

Question

I am given a recursion: $$(n+1)b_{n+1} = b_n + \frac{1}{(n-1)!}$$ we are to find a substitution which will convey our equation to a linear form and of course we are to find $b_{n}$.
I have no idea what the substitution should look like. I would appreciate any help.

Answer 1

Hint. Consider the substitution $u_n=n!b_n$ then recursion becomes $$u_{n+1}=u_n+n.$$

Answer 2

Further hint (with respect to Robert Z answer). Once you have $u_{n+1}=u_n+n$ you also have

$$ u_N-u_0=\sum_{n=0}^{N-1}\left(u_{n+1}-u_n\right) = \sum_{n=0}^{N-1} n = \frac{N(N-1)}{2} $$ from which $u_N = \frac{N(N-1)}{2}+u_0$ follows.