An article introducing continued fractions with some simple puzzles for the reader.
For which values of n is the Fibonacci number fn even? Which
Fibonnaci numbers are divisible by 3?
When is a Fibonacci sequence also a geometric sequence? When the
ratio of successive terms is the golden ratio!
You will probably need to experiment a little to try to form a conjecture for the form of the coefficients. You can then use induction to try to prove your conjecture.