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Golden Ratio

Age 16 to 18 Challenge Level:

The 'divina proporzione' or golden ratio, represented by the Greek letter phi, is derived from the equation below where $a$ and $b$ are parts of a line.

$a+b:a=a:b$

i.e. $ \frac{a+b}{a}=\frac{a}{b}=\Phi\ \quad $(phi)

If $b = 1$ show that $\Phi = a = (\sqrt 5 + 1 )/2 = 1.618034...$.

In the following equation what does $x$ equal?

$$\Phi^{\left(\Phi^x-\frac{x-1}{\Phi}\right)}-\frac{1}{\Phi}=x$$