
Remember that the Fibonacci sequence is
1, 1, 2, 3, 5, 8, 13, 21, ...
and the rule to generate the next term is to add the two previous
terms.
Let's look at the sequence of continued fractions generated by
The first term is 1. The second term is 1+1/1 or 2. The third term is
is 1 + 1/(1+1) or 1+1/2, which is 3/2. The
fourth term is 1 + 1/[1 + 1/(1 + 1/1)], which is 1 + 1/[3/2], or 1 + 2/3,
which is 5/3. Let's try that sequence down:
1/1, 2/1, 3/2, 5/3, ...
Can you see it's connection to the Fibonacci sequence now?
The sequence generated by this particular continued fraction generates the ratios
of succesive terms of the Fibonacci sequence.
