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?



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