Prove that $ (\frac{1+\sqrt5}{2})^{n}+(\frac{1+\sqrt5}{2})^{n-1} = (\frac{1+\sqrt5}{2})^{n+1}$.

Question

I have to get from $ (\frac{1+\sqrt5}{2})^{n}+(\frac{1+\sqrt5}{2})^{n-1}$ to $(\frac{1+\sqrt5}{2})^{n+1}$ however I do not know how to get there since i do not know what to do with the exponents.

(not sure if I used the right tag)

Answer 1

HINT: Let $\varphi=\frac12\left(1+\sqrt5\right)$; you have to show that $\varphi^n+\varphi^{n-1}=\varphi^{n+1}$. Divide through by $\varphi^{n-1}$ to see that all you really need to show is that $\varphi+1=\varphi^2$. That’s a matter of fairly straightforward arithmetic.