Here, $\phi$ is the golden ratio and $f_n$ is the $n^{th}$ Fibonacci number. The formula I'm using is actually the closed form of the Lucas numbers.
Let $n = 1$. Then $f_n = 1$ and $\phi + 1 - \phi = 1$. Thus, the case holds where $n = 1$, establishing our base case.
Assume the case holds for all natural numbers less than or equal to $k$. Then $f_{n+1} = f_n + f_{n-1} = \phi^k + (1-\phi)^k + \phi^{k-1} + (1-\phi)^{k-1} = \phi^{k-1}(1 + \phi) + (1-\phi)^{k-1}(2-\phi) = \phi^{k-1}\phi^2 + (1 - \phi)^{k-1}(1-\phi)^2 = \phi^{k+1} + (1-\phi)^{k+1}$. Thus the case holds where $n = k+1$. By mathematical induction, the claim holds for all $n \in N$.
However, $f_3 =2$ and $\phi^3 + (1-\phi)^3 = 4$. Why is this?