Maximal length of a finite real sequence Suppose that $(a_n)_{1 \le n \le N}$ is a finite sequence of reals such that the sum of any 7 consecutive terms is (strictly) negative and the sum of any 11 consecutive terms is (strictly) positive.
What is the maximal length of this finite sequence of reals?
I tried creating finite sequences having the given property. I succeeded in creating a sequence of 16 terms. Namely $5,5,-13,5,5,5,-13,5,5,-13,5,5,5,-13,5,5$... but wasn't able to do more.

$x_1+\dots+x_7<0$ and $x_8+\dots+x_{14}<0$, so $x_1+\dots+x_{14}<0$. But $x_4+\dots+x_{14}>0$, so $x_1+x_2+x_3<0$.
$x_5+\dots+x_{11}<0$ and $x_1+\dots+x_{11}>0$, so $x_4>0$. From there is it easy to get a contradiction if you have 17 terms.
