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Fibonacci Surprises

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

You may have seen this sequence before: $$1, 1, 2, 3, 5, 8, 13, 21, ... $$

It is called the Fibonacci Sequence, and each term is calculated by adding together the previous two terms in the sequence.

Try adding together any three consecutive Fibonacci numbers.
What do you notice?
Can you explain it?


Choose any four consecutive Fibonacci numbers. Add the first and last, and divide by two.
What do you notice?
Can you explain it?


Add together any six consecutive Fibonacci numbers and divide by four.
What do you notice?
Can you explain it?


Can you discover any Fibonacci Surprises of your own?


With thanks to Don Steward, whose ideas formed the basis of this problem.