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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,

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  • $\begingroup$ You will be perhaps interested by the so-called "metallic sequences" considered here $\endgroup$
    – Jean Marie
    Commented May 1, 2022 at 21:16
  • $\begingroup$ Thanks, but I am not sure that it answers my question unless I am missing something. $\endgroup$
    – Mike
    Commented May 2, 2022 at 18:59

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