The inductive step is \(\frac{1}{\sqrt{5}}\phi^{n-1}+\frac{1}{\sqrt{5}}\phi^{n-2}=\frac{1}{\sqrt{5}}\phi^{n-2}(\phi-1)=\frac{1}{\sqrt{5}}\phi^n\), because \(\phi^2=\phi+1\), and all that is true for \(\psi\) too. The formula in the book is correct. \(\psi<0\), so its integer powers are very small for large \(n\).