# 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 $S(6+4k)$Â  is an integer multiple of $F(5+2k)$.

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

### 1 Response to "Fibonacci tricks from Palm Breeze CAFE"

[…] Our thanks to the Republic of Maths blog that brought this to our attention, via this video link. […]