Suppose we have an infinite sequence
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and that for all we have
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(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
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(2) |
Substituting the expression given by this equation for the in the denominator on the right hand side of the same expression gives
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Another substitution gives
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Continuing on in this vein, it is not too hard to prove that is equal to the infinite continued fraction
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(You can find out more about continued fractions here.)
Now back to our sequence. The ratio of successive terms is
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By equation (1) this is equal to
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Now
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Applying equation (1) to the term we get
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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
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Letting tend to infinity we can then prove that the ratio of successive terms in the sequence converges to infinite continued fraction
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Which, as we have sketch-proved above, is equal to