Proving that $\sqrt{\sqrt{\sqrt{\sqrt{\frac{1+\sqrt{5}}{2}+1}+1}+1}+1} = \phi$

Question

I have this statement:

Prove that $\sqrt{\sqrt{\sqrt{\sqrt{\frac{1+\sqrt{5}}{2}+1}+1}+1}+1} = \phi$

I have tried to develop algebraically, without having anything concrete. I have only obtained the longest problem, since I have tried to raise to 2 to eliminate roots.

So, how can I prove this? Thanks in advance.

Answer 1

I presume $\phi=\frac{1+\sqrt5}2$. Can you prove that $\sqrt{1+\phi}=\phi$? If so then $$\phi=\sqrt{\phi+1}=\sqrt{\sqrt{\phi+1}+1}=\sqrt{\sqrt{\sqrt{\phi+1}+1}+1}= \cdots$$ ad nausueam.