### Golden Powers

You add 1 to the golden ratio to get its square. How do you find higher powers?

### Continued Fractions I

An article introducing continued fractions with some simple puzzles for the reader.

### Fibonacci Factors

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

# Pythagorean Fibs

##### Stage: 5 Challenge Level:

Parts (2) and (3) use the earlier parts of the question.

For part (3), as usual try small values of $n$ first, look for a pattern and make a conjecture about the result you expect might always be true.

To prove your conjecture take $$F_n=b-a,\ F_{n+1}=a,\ F_{n+2}= b,\ F_{n+3}= b+a$$ because this symmetry in the algebra will make the working simpler.