Everyone knows that we can obtain the golden ratio from the following proportion:
$$\frac{a}{b} = \frac{a+b}{a}$$
We also know that we get ${\phi}^N$ when we try to find a function that satisfies the following Fibonacci sequence characteristic: $$A^{N+2}=A^{N+1} + A^N$$
Algebraically, I know why they are related(both the ratio $\frac{a}{b}$ in the first eq and the constant $A$ in the second eq are solutions to the same second degree equation: $x²=x+1$). However, I have absolutely no intution for why that would be the case. Thanks in advance for any help,
