Given $\displaystyle L_{1}=1,L_{2}=3,L_{n+2}=L_{n}+L_{n+1}$, how can I prove that $\displaystyle L_{3}=\alpha^3-\alpha^{-3}$ where $\displaystyle \alpha=\frac{1+\sqrt{5}}{2}$??
If you are supposed to find $\displaystyle L_3$, then just check if the equality hold or not.
If you are supposed to find the formulla of $\displaystyle L_n$, the characteristic equation $\displaystyle x^2-x-1=0$(can you find the roots)will help you.