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'Golden Ratio' printed from https://nrich.maths.org/
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.
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$a+b:a=a:b$
i.e. $ \frac{a+b}{a}=\frac{a}{b}=\Phi\ \quad $(phi)
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If $b = 1$ show that
$\Phi = a = (\sqrt 5 + 1 )/2 = 1.618034...$.
In the following equation
what does $x$ equal?
$$\Phi^{\left(\Phi^x-\frac{x-1}{\Phi}\right)}-\frac{1}{\Phi}=x$$