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?
by
Guillermo Templado