TITLE: The sum of 8 consecutive Fibonacci numbers is not a Fibonacci number
QUESTION [7 upvotes]: While studying Fibonacci numbers, I came up with this problem. Of course $F_n = F_{n-1}+F_{n-2}$. I'm sort of stuck with first realizing how to show a number actually isn't a Fibonacci number. I thought that I could somehow rewrite the sum $$F_n+\cdots+F_{n+7}$$
into some sort of rearrangement of $F_n$'s and $F_{n+1}$'s. Could anyone help me show that the sum of 8 consecutive Fibonacci numbers is not a Fibonacci number?
Thanks

REPLY [7 votes]: We have that 
$$S:=F_n+\dots+F_{n+7}=F_{n+2}+F_{n+4}+F_{n+6}+F_{n+8}>F_{n+8}.$$
But on the other hand
$$F_{n+9}=F_{n+8}+F_{n+7}=F_{n+8}+F_{n+6}+F_{n+5}=\dots\\=F_{n+8}+F_{n+6}+F_{n+4}+F_{n+2}+F_{n}+F_{n-1}=S+F_{n-1}>S.$$
In sum, this means that $F_{n+8}<S<F_{n+9}$.
But the Fibonacci sequence is streactly increasing, so there is no Fibonacci number between $F_{n+8}$ and $F_{n+9}$, i.e. $S$ is no Fibonacci number.