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interesting fact about golden mean

i was solving some problem, and i found a fun facts about the golden mean. \( Golden\quad mean =\Phi=\dfrac{1+\sqrt{5}}{2}\)\[\] \[ F(n)={(((((\Phi-1)^2-2)^2-2)^2-2\dotsm}[n\quad times]\] \[ f(n)=\begin{cases} -\Phi\quad if \quad n\in [odd]\\ \Phi-1\quad if\quad n\in[even]\\ \end{cases}\] seems interesting, but why??? \[\] note that \[\Phi^2 =\Phi+1\] now, \[(\Phi-1)^2-2= \Phi^2-2\Phi+1-2=\Phi+1-2\Phi-1=\boxed{-\Phi}\] \[((\Phi-1)^2-2)^2-2=(-\Phi)^2 -2=-\Phi^2-2=-\Phi+1-2=\boxed{\Phi-1}\] \[(((-\Phi-1)^2-2)^2-2)^2 -2=(\Phi-1)^2-2=\boxed{-\Phi}\] and on and on and on, this is quite fascinating because \(\Phi\) is involved in a recurring pattern!!!

Note by Aareyan Manzoor
2 years ago

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