The golden ratio

Algebra Level 3

The golden ratio ϕ\phi is defined to be the positive root of x2x1=0x^2 - x - 1 = 0.

1)ϕ=1+1+1+1)\phi = \sqrt {1 + \sqrt {1 +\sqrt {1 + \ldots } }}

2)ϕ=1+11+11+2) \phi = 1 + \frac {1}{1 + \frac {1}{1 + \ldots } }

3)3) If FnF_{n} represents the Fibonacci's sequence then ϕ=limnFn+1Fn\phi =\displaystyle \lim_{n \to \infty} \frac {F_{n+1}}{F_{n}}

4)ϕ4)\phi implicitly appears in numerous works of art.

How many statments above are true?

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