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NRICH: Fibonacci Surprises

Play around with the Fibonacci sequence and discover some surprising results!

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.

Age 11 to 14

Math Topics
Algebra & Pre-Algebra, Patterning & Sequencing, Quadratic Equations
Middle School, High School, Educator
6th Grade, 7th Grade, 8th Grade, 9th Grade

## Organization

NRICH (University of Cambridge)

Challenge