Metallic means (metallic ratios), also called silver means (silver ratios), are a generalization of the golden ratio.
The more general simple continued fraction expressions
are known as the silver means
or metallic means
of the successive natural numbers
. The golden ratio is the silver mean between
, while the silver ratio is the silver mean between
. The term "bronze ratio
", or terms using other names of metals, are occasionally used to name subsequent silver means.
The values of the first ten silver means are shown at right.
Notice that each silver mean is a root of the simple quadratic equation
is any positive natural number. Hence
which leads to the above continued fraction.
The silver mean of
is also given by the integral