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Golden Ratio Values

The Golden Ratio:

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

$${\mathbf n}$$	$${\boldsymbol{\phi^n}}$$	$${\boldsymbol{\frac{1}{\phi^n}}}$$	$${\boldsymbol{\phi^n + \frac{1}{\phi^n}}}$$	$${\boldsymbol{\phi^n - \frac{1}{\phi^n}}}$$
1	$${\frac{1 + \sqrt{5}}{2}}$$	$${\frac{\sqrt{5} - 1}{2}}$$	$${\sqrt{5}}$$	$${1}$$
2	$${\phi + 1 = \frac{3 + \sqrt{5}}{2} }$$	$${1 - \frac{1}{\phi} = \frac{3 - \sqrt{5}}{2}}$$	$${3}$$	$${\sqrt{5}}$$
3	$${\frac{4 + 2\sqrt{5}}{2}}$$	$${\frac{2\sqrt{5} - 4}{2}}$$	$${2\sqrt{5}}$$	$${4}$$
4	$${\frac{7 + 3\sqrt{5}}{2}}$$	$${\frac{7 - 3\sqrt{5}}{2}}$$	$${7}$$	$${3\sqrt{5}}$$
5	$${\frac{11 + 5\sqrt{5}}{2}}$$	$${\frac{5\sqrt{5} - 11}{2}}$$	$${5\sqrt{5}}$$	$${11}$$
6	$${\frac{18 + 8\sqrt{5}}{2}}$$	$${\frac{18 - 8\sqrt{5}}{2}}$$	$${18}$$	$${8\sqrt{5}}$$
7	$${\frac{29 + 13\sqrt{5}}{2}}$$	$${\frac{13\sqrt{5} - 29}{2}}$$	$${13\sqrt{5}}$$	$${29}$$
8	$${\frac{47 + 21\sqrt{5}}{2}}$$	$${\frac{47 - 21\sqrt{5}}{2}}$$	$${47}$$	$${21\sqrt{5}}$$

Formulas and Links:

\({\phi = \frac{1 + \sqrt{5}}{2}}\)

\({\phi^2 = \phi + 1}\)

\({1 - \phi = - \frac{1}{\phi}}\)

\({\phi - \frac{1}{\phi} = 1}\)

\({1 - \frac{1}{\phi} = \frac{1}{\phi^2}}\)

Golden Ratio Quiz