Solution
= Solution
The <linearity> of the <Fibonacci recurrence> allows a <linear combination>
$$
F_n=A\phi^n+B\psi^n.
$$
The <initial conditions> give $A+B=0$ and $A\phi+B\psi=1$. Since $\phi-\psi=\sqrt5$, we obtain $A=1/\sqrt5$ and $B=-1/\sqrt5$. Thus <Binet's formula> is
$$
\boxed{F_n=\frac{\phi^n-\psi^n}{\sqrt5}
=\frac1{\sqrt5}\left[\left(\frac{1+\sqrt5}{2}\right)^n-
\left(\frac{1-\sqrt5}{2}\right)^n\right].}
$$