Euler-Binet Formula in another form

Question

So I tried deriving the Euler-Binet formula without induction, and I ended up with this:

$$F(n) = \frac{\phi^{n+1}}{\phi^{2}+1} + \frac{1}{(-\phi)^{n-1}(\phi^{2}+1)}$$

How do I deduce the Euler-Binet from this form?

Answer 1

You have $$F(n) = \frac{\phi\cdot \phi^n}{\phi^2+1} + \frac{-\phi}{(-\phi)^n(\phi^2+1)} = \frac{\phi}{\phi^2+1} \left( \phi^n - (-\phi)^{-n} \right).$$ You can easily verify that $$\frac{\phi}{\phi^2+1} = \frac{\dfrac{1+\sqrt 5}{2}}{\dfrac{5 + \sqrt{5}}{2}} = \frac{1 + \sqrt{5}}{5 + \sqrt 5} = \frac 1{\sqrt 5}$$ so that $$F(n) = \frac{1}{\sqrt 5} \left( \phi^n - (-\phi)^{-n} \right)$$ which is the Euler-Binet formula.