Solution
= Solution
Use <mathematical induction> with two base cases. At $n=0$, the formula gives $(1-1)/\sqrt5=0=F_0$. At $n=1$, it gives $(\phi-\psi)/\sqrt5=1=F_1$.
If it holds at $n-1$ and $n$, then the <Fibonacci recurrence> gives
$$
F_{n+1}=\frac{\phi^n+\phi^{n-1}-\psi^n-\psi^{n-1}}{\sqrt5}.
$$
The identities $\phi^2=\phi+1$ and $\psi^2=\psi+1$ turn the numerator into $\phi^{n+1}-\psi^{n+1}$. Thus
$$
\boxed{F_n=(\phi^n-\psi^n)/\sqrt5\quad\text{for every }n\ge0.}
$$