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**Metallic means** (**metallic ratios**), also called **silver means** (**silver ratios**), are a generalization of the golden ratio.

The more general simple continued fraction expressions

- $n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{\ddots }}}}}}}}}}=[n;n,n,n,n,\dots ]={\frac {1}{2}}\left(n+{\sqrt {n^{2}+4}}\right)\,$

are known as the

**silver means** or

**metallic means**^{[1]} of the successive

natural numbers. The golden ratio is the silver mean between

and

, while the silver ratio is the silver mean between

and

. The term "

**bronze ratio**", or terms using other names of metals, are occasionally used to name subsequent silver means.

^{[1]} The values of the first ten silver means are shown at right.

^{[2]} Notice that each silver mean is a root of the simple quadratic equation

- $x^{2}-nx-1=0,\,$

where

is any positive natural number. Hence

- $x-n={\frac {1}{x}}={\frac {1}{n+(x-n)}},\,$

which leads to the above continued fraction.

The silver mean of

is also given by the integral

- $S_{n}=\int _{0}^{n}{\left({x \over {2{\sqrt {x^{2}+4}}}}+{{n+2} \over {2n}}\right)}\,{\rm {d}}x.$

## Notes