The solutions of the characteristic equation^{} ${x}^{2}x1=0$ are

$$\varphi =\frac{1+\sqrt{5}}{2},\psi =\frac{1\sqrt{5}}{2}$$ 

so the closed formula for the Fibonacci sequence^{} must be of the form

$${f}_{n}=u{\varphi}^{n}+v{\psi}^{n}$$ 

for some real numbers $u,v$. Now we use the boundary conditions of the recurrence, that is, ${f}_{0}=0,{f}_{1}=1$, which means we have to solve the system

$$0=u{\varphi}^{0}+v{\psi}^{0},1=u{\varphi}^{1}+v{\psi}^{1}$$ 

The first equation simplifies to $u=v$ and substituting into the second one gives:

$$1=u\left(\frac{1+\sqrt{5}}{2}\right)u\left(\frac{1\sqrt{5}}{2}\right)=u\left(\frac{2\sqrt{5}}{2}\right)=u\sqrt{5}.$$ 
