Something interesting about Fibonacci numbers

Maybe everyone else knows this already – but I did not until I was reading Structure and Interpretation of Computer Programs and I picked it up…

Well, the Fibonacci series are defined as:

Fibonacci series definitionThis generates the sequence 0, 1, 1, 2, 3, 5 etc.

(Apparently Fibonacci, real name Leonardo Pisano Bigollo, used the sequence to model rabbit populations in the 13th century, though perhaps more importantly he essentially codified the western system of arabic numbers. The sequence was known by Indian mathematicians long before then though.)

But the sequence is perhaps most interesting for its relationship to the “golden ratio” (or golden mean), phi:

Golden meanwhich means phi has the value:

phi

This is the ratio where:

Golden ratio

 

The interesting thing about the Fibonacci series for n is that each number in the series is the closest integer to:

SeriesProof

 

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