# Nonlinear Difference Equation Solved by Substitution

I'm reading Introduction to Dynamic Systems by David Luenberger. Question 2.1 asks to solve a nonlinear difference equation $$y(k+1) = \frac{y(k)}{b + y(k)} \quad \text{for } k=1, 2, \ldots$$ He recommends using a change of variables to convert into a linear difference equation and then solve.

Writing out $$y(k+3)$$ shows a pattern $$y(k+3) = \frac{y(k)}{b^3 + (b^2 + b + 1) y(k)}.$$ The denominator is a geometric sequence $$1 + b + b^2 + \cdots + b^{k-1} = \frac{1-b^k}{1-b}$$ with an identity that can be substituted and written out in general as $$y(k+1) = \frac{y(k)}{b^k + \frac{1-b^k}{1-b} y(k)}.$$ However I do not know how to find the solution of this problem either.

Would anyone have an idea how to approach this problem?