Republic of Mathematics blog

Fibonacci tricks from Palm Breeze CAFE

Posted by: Gary Ernest Davis on: November 17, 2010

Here’s the algebra:

x

x

x

x

x

x

x

More generally, if F(n) is the n^{th} Fibonacci number, where F(1)=A, F(2)=B \textrm{ and } F(n+2)=F(n+1)+F(n), and S(n) is the sum of the Fibonacci numbers F(1)+\ldots+F(n) then S(6+4k)  is an integer multiple of F(5+2k).

For example, S(14)=F(1)+\ldots+F(14)=29\times F(9), and S(98)=F(1)+\ldots+F(98)=17393796001\times F(51).

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