User:Andrewdavinci

$varphi=$ 1.61803398874989484820458683436563811772 0309179805762862135448622705260462818902 44970720720418939113748475

Golden triangle

A Fibonacci spiral that approximates the golden spiral, using Fibonacci sequence square sizes up to 34.

$\varphi ={\sqrt[{3}]{(}}\ 2+{\sqrt {5}})$

$\varphi -1={\tfrac {1}{\varphi }}$.

${\tfrac {1}{{\sqrt[{3}]{(}}\ 2+{\sqrt {5}}}}={\sqrt[{3}]{(}}\ 2+{\sqrt {5}})-4$

$\varphi$² = $\varphi +1$

$\varphi ={\frac {1+{\sqrt {5}}}{2}}$

$\varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}\,.$

$-{\frac {\varphi }{2}}=\sin 666^{\circ }=\cos(6\cdot 6\cdot 6^{\circ }).{}$

$\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }$

$\varphi ={1 \over 2}\csc(\pi /10)={1 \over 2}\csc 18^{\circ }$

$\varphi =2\cos(\pi /5)=2\cos 36^{\circ }.\,$

$\sum _{n=1}^{\infty }|F(n)\varphi -F(n+1)|=\varphi \,.$

$\varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}\,.$

${\begin{aligned}3\varphi ^{3}-5\varphi ^{2}+4&=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4\\&=3[(\varphi +1)+\varphi ]-5(\varphi +1)+4\\&=\varphi +2\approx 3.618.\end{aligned}}$

The golden ratio's decimal expansion can be calculated directly from the expression

$\varphi ={1+{\sqrt {5}} \over 2},$

√5 ≈ 2.2360679774997896964.

$x_{n+1}={\frac {(x_{n}+5/x_{n})}{2}}$

x_n and x_n−1.

$x_{n+1}={\frac {x_{n}^{2}+1}{2x_{n}-1}},$

x − 1 − 1/x = 0,

$x_{n+1}={\frac {x_{n}^{2}+2x_{n}}{x_{n}^{2}+1}}.$

F₂₅₀₀₁ and F₂₅₀₀₀,