Suppose we have an infinite sequence

and that for all we have

(1) |

where is a positive integer.

In the main article we claimed that in this case the ratio of successive terms of the sequence converges to the metallic mean . We’ll now give you a justification of this claim. Recall that

(2) |

Substituting the expression given by this equation for the in the denominator on the right hand side of the same expression gives

Another substitution gives

Continuing on in this vein, it is not too hard to prove that is equal to the infinite *continued fraction*

(You can find out more about continued fractions here.)

Now back to our sequence. The ratio of successive terms is

By equation (1) this is equal to

Now

Applying equation (1) to the term we get

You can see the picture: continuing to make substitutions according to equation (1), we end up with a finite continued fraction expression for of the form

Letting tend to infinity we can then prove that the ratio of successive terms in the sequence converges to infinite *continued fraction*

Which, as we have sketch-proved above, is equal to