Find the rational numbers, in lowest terms, given by each of the following continued fractions. Do you notice anything interesting? What value would have a simple continued fraction representation with an infinite string of $1$'s (i.e., $[1,\overline{1}]$)? Knowing that the continued fractions given below should better and better approximate $[1,\overline{1}]$, what might one conclude?

  1. [1;1]
  2. [1;1,1]
  3. [1;1,1,1]
  4. [1;1,1,1,1]

After calculating the first several convergent values, it shouldn't be hard to see the pattern...

$$\begin{array}{rcl} [1;1] &=& 2\\ [1;1,1] &=& 3/2\\ [1;1,1,1] &=& 5/3\\ [1;1,1,1,1] &=& 8/5\\ [1;1,1,1,1,1] &=& 13/8\\ \vdots \end{array}$$

We are looking at ratios of successive Fibonacci numbers!

Of course, we expect the convergents to get closer and closer in value to $[1,\overline{1}]$, which we can compute in the standard way. $$[1,\overline{1}] = \frac{1+\sqrt{5}}{2}$$

The observant among you might recognize this value as the golden ratio, $\varphi$.

As such, we conclude that the limiting quotient of successive Fibonacci numbers must be the golden ratio. That ties some nice things together, doesn't it!