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?

4 0 2 3 1

1 solution

Akshat Sharda
Nov 23, 2015

All the above statements are true.

$1)\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{\ldots}}}}=x \\ \sqrt{1+x}=x\Rightarrow x+1=x^2 \\ x^2-x-1=0\Rightarrow x=\phi$

$2) 1 + \frac {1}{1 + \frac {1}{1 + \ldots } }=x \\ 1+\frac{1}{x}=x \Rightarrow x+1=x^2 \\ x^2-x-1=0\Rightarrow x=\phi$

$3)$ For proof, see here .

$4)$ It is a fact.

Please,could you prove 3)?, the page is not still created.

Guillermo Templado - 5 years, 6 months ago

Now see the link .

Akshat Sharda - 5 years, 6 months ago

Right. For seeing the 4º point,we can cite that the face of Parthenon in Greece and Gioconda picture keeps this proportion, for example. Thank you very much for your proof.

Guillermo Templado - 5 years, 6 months ago

@Guillermo Templado – Studies show that even our face has the same proportion.

Akshat Sharda - 5 years, 6 months ago

@Akshat Sharda – Right. We can find the golden ratio in Nature,too.

Guillermo Templado - 5 years, 6 months ago

I think 4 is debatable. They might use approximations of phi in art, but I am not convinced that they are typically aiming for it.

Michael Esplin - 5 years, 6 months ago

The golden ratio

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