Binet formula for the Fibonacci numbers

The Fibonacci numbers satisfy $F_0=0$, $F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$. Find an explicit formula for $F_n$.

$F_n=\dfrac{\varphi^n-\psi^n}{\sqrt5}$ with $\varphi=\tfrac{1+\sqrt5}{2}$, $\psi=\tfrac{1-\sqrt5}{2}$. Try $F_n=r^n$: the recursion gives $r^2=r+1$, whose roots are $\varphi,\psi$. The general solution $A\varphi^n+B\psi^n$ fitted to $F_0=0,F_1=1$ gives $A=-B=1/\sqrt5$.