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

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Continued Fractions I

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

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Fibonacci Factors

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

Plus or Minus

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
This problem complements the material in the article The Golden Ratio, Fibonacci Numbers and Continued Fractions.

For a sequence of, mainly more elementary, problems on these topics see Golden Mathematics.