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March 5th, 2017, 11:28 AM   #1
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Math Focus: Trigonometry and complex numbers
Why does this approach the golden ratio?

Hi. If you pick any two number, say for instance 5 and 7 and you keep adding them together (much like the Fibonacci Sequence) you get this:

$\displaystyle 5, 7, 12, 19, 31, 50, 81, 131, ...$ and so on.

The ratio between the last two numbers always approaches the golden ratio, no matter which set of numbers you choose. But I just cannot understand why this happens... Can anybody provide some good sources and/or explanation?

Thanks in advance,

Xxmarijnw
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March 5th, 2017, 12:21 PM   #2
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Suppose we look at the recurrence:

$\displaystyle A_{n}=A_{n-1}+A_{n-2}$

The characteristic equation is:

$\displaystyle r^2-r-1=0$

which has roots:

$\displaystyle r=\frac{1\pm\sqrt{5}}{2}$

And so the closed form is:

$\displaystyle A_{n}=k_1\left(\frac{1+\sqrt{5}}{2}\right)^n+ k_1\left(\frac{1-\sqrt{5}}{2}\right)^n$

The parameters $k_i$ will depend in the initial values.

Since:

$\displaystyle \left|\frac{1-\sqrt{5}}{2}\right|<1$

Then as $n$ grows without bound, we will find:

$\displaystyle A_n\to k_1\left(\frac{1+\sqrt{5}}{2}\right)^n$

And so:

$\displaystyle \frac{A_{n+1}}{A_{n}}\to \frac{k_1\left(\dfrac{1+\sqrt{5}}{2}\right)^{n+1}} {k_1\left(\dfrac{1+\sqrt{5}}{2}\right)^n}= \frac{1+\sqrt{5}}{2}=\varphi$
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