Let m, k, and c be positive integers with k>c, and let \lambda be a positive, non-integer real root of the equation \lambda^{m+1}-k \lambda^m-c=0. Let f: \mathbb{Z}^{+} \rightarrow \mathbb{Z} be defined by f(n)=\lfloor\lambda n\rfloor for all n \in \mathbb{Z}^{+}. Show that f^{m+1}(n) \equiv c n-1(\bmod~ k) for all n \in \mathbb{Z}^{+}. (Here, $\mathbb{Z}^{+}denotes the set of positive integers, \lfloor x\rfloor$ denotes the greatest integer less than or equal to x, and f^{m+1}(n)=f(f(\ldots f(n) \ldots)) where f appears m+1 times.)