# 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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