Prove that $$a_n = \frac{1}{\sqrt{5}}\left((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n\right)$$ if $a_{n+1} = a_{n-1} + a_n$
My idea to solve this was to do what follows:
(1) Check if this relation holds for $n =1$ and $n=2$
(2) Assume that the formula is true for $n$ and $n-1$
(3) Evaluate $a_{n+1}=a_n+a_{n-1}$, substitute for the right hand side and try to to get
$$a_{n+1} = \frac{1}{\sqrt{5}}\left((\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1}\right)$$
After I put some work in, I carried it to this form:
$$\frac{1}{\sqrt{5}}\left(\left( \frac{1+\sqrt{5}}{2} \right)^{n-1} \left( \frac{3+\sqrt{5}}{2} \right) -\left(\frac{1-\sqrt{5}}{2} \right)^{n-1} \left( \frac{3-\sqrt{5}}{2}\right)\right)$$
I have checked and $\frac{3+\sqrt{5}}{2}=(\frac{1+\sqrt{5}}{2})^2$
Is this proof formally valid?
