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.

score 6 · Answer 1 · answered Aug 18 '18 at 02:21

6

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.

answered Aug 18 '18 at 02:21

Angina Seng

158,341

This is the first time I have seen "ad nausueam." What does it define and when is it used? Is is like QED? – For the love of maths Aug 18 '18 at 04:31
Well, that's easy to prove. But how can I be sure that it will be true if I add more roots? – ESCM Aug 19 '18 at 20:25
Is a inductive proof? – ESCM Aug 21 '18 at 20:40

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

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