A brief glimpse into the past

Metallic mean

The metallic means of the successive natural numbers are the continued fractions: n + 1 n + 1 n + 1 n + 1 n + ⋱ = [ n ; n , n , n , n , … ] = n + n 2 + 4 2 . {\displaystyle n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+\ddots \,}}}}}}}}=[n;n,n,n,n,\dots ]={\frac {n+{\sqrt {n^{2}+4}}}{2}}.} 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.