{"document_url": "https://en.wikipedia.org/wiki/Metallic_mean"}
Golden ratio within the pentagram and silver ratio within the octagon.

The metallic means (also ratios or constants) of the successive natural numbers are the continued fractions:
The golden ratio (1.618...) is the metallic mean between 1 and 2, while the silver ratio (2.414...) is the metallic mean between 2 and 3. The term "bronze ratio" (3.303...), or terms using other names of metals (such as copper or nickel), are occasionally used to name subsequent metallic means. The values of the first ten metallic means are shown at right. Notice that each metallic mean is a root of the simple quadratic equation:
, where
is any positive natural number.

As the golden ratio is connected to the pentagon (first diagonal/side), the silver ratio is connected to the octagon (second diagonal/side). As the golden ratio is connected to the Fibonacci numbers, the silver ratio is connected to the Pell numbers, and the bronze ratio is connected to OEIS: A006190. Each Fibonacci number is the sum of the previous number times one plus the number before that, each Pell number is the sum of the previous number times two and the one before that, and each "bronze Fibonacci number" is the sum of the previous number times three plus the number before that. Taking successive Fibonacci numbers as ratios, these ratios approach the golden mean, the Pell number ratios approach the silver mean, and the "bronze Fibonacci number" ratios approach the bronze mean.

Properties
Gold, silver, and bronze ratios within their respective rectangles.

These properties are valid only for integers m. For nonintegers the properties are similar but slightly different.

The above property for the powers of the silver ratio is a consequence of a property of the powers of silver means. For the silver mean S of m, the property can be generalized as
where
Using the initial conditions K0 = 1 and K1 = m, this recurrence relation becomes
The powers of silver means have other interesting properties:

If n is a positive even integer:
Additionally,
Also,
In general:
The silver mean S of m also has the property that
meaning that the inverse of a silver mean has the same decimal part as the corresponding silver mean.
where a is the integer part of S and b is the decimal part of S, then the following property is true:
Because (for all m greater than 0), the integer part of Sm = m, a = m. For m > 1, we then have
Therefore, the silver mean of m is a solution of the equation
It may also be useful to note that the silver mean S of −m is the inverse of the silver mean S of m
Another interesting result can be obtained by slightly changing the formula of the silver mean. If we consider a number
then the following properties are true:
if c is real,
if c is a multiple of i.

The silver mean of m is also given by the integral
Another interesting form of the metallic mean is given by
Trigonometric expressions
Geometric Construction

The metallic mean for any given integer
can be constructed geometrically in the following way. Define a right triangle with sides
and
having lengths of
and
, respectively. The
th metallic mean
is simply the sum of the length of
and the hypotenuse,
.

For
,
and so
φ.

Setting
yields the silver ratio.
Thus
Likewise, the bronze ratio would be calculated with
so
yields
Non-integer arguments sometimes produce triangles with a mean that is itself an integer. Examples include N = 1.5, where
and
Which is simply a scaled-down version of the 3-4-5 Pythagorean triangle.