The golden ratio

Algebra Level 3

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

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

\(2) \phi = 1 + \frac {1}{1 + \frac {1}{1 + \ldots } }\)

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

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

How many statments above are true?

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