Why does the golden ratio satisfy $\phi = 1 + \frac{1}{\phi}?$

Question

I want to show this result starting from the property that $\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}$.

I presume that to get $\phi = \frac{1}{\phi} $ I need to find the limit of the equation above.

However, I'm unsure how to do this. It's easy to see where the "1" comes from, but I'm not sure about the $\frac{1}{\phi}$ part.

Answer 1

By inspection: $\;\displaystyle\phi = \color{red}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}} = \color{blue}1 + \frac{1}{\color{red}{1 + \frac{1}{1 + \frac{1}{...}}}} = \color{blue}{1}+\frac{1}{\color{red}{\phi}}\,$.

For a formal proof, consider the recurrence $\;\displaystyle\phi_0=1, \;\phi_{n+1}=1+\frac{1}{\phi_n}\,$ and define $\,\phi\,$ as its limit.