Define three sequences $x_n, y_n, z_n$ for $n=1, 2, \dots, $ by $x_1 = 2$, $y_1 = 4$, $z_1 = \frac{6}{7}$ and the recursion $$ x_{n+1} = \frac{2x_n}{x_n^2-1}, \quad y_{n+1} = \frac{2y_n}{y_n^2-1}, \quad z_{n+1} = \frac{2z_n}{z_n^2-1} $$
Is it possible to have $x_n + y_n + z_n = 0$ for some $n$?
Looking at each sequence separately, they look quite messy and are not monotone and also don't seem to converge to a limit. I have tried an induction method, to start from $x_{n+1} + y_{n+1} + z_{n+1}$, expand out using the recursion, and try to show that it is non-zero provided that $x_n+y_n+z_n$ is non-zero, but I am not getting anywhere because it is too messy. Maybe it can be simplified to showing that $x_n + y_n > z_n$ for all $n$ or something along those lines which will be sufficient to prove the result.
Alternatively, I think it would be neater to find some invariant but I am really bad at finding invariants and I don't see any clue on how to start. The most obvious $x_n +y_n + z_n$ is not constant so that doesn't work.
Another thing maybe is to look at the sequence of numerators and denominators i.e. let $a_n/b_n = x_n$ where $a_n$ and $b_n$ are coprime and work from there but it also gets messy very quickly. I have $\frac{a_{n+1}}{b_{n+1}} = \frac{2a_nb_n}{a_n^2 - b_n^2} = \frac{\sqrt{a_n^2b_n^2}}{\frac{a_n^2-b_n^2}{2}}$ which doesn't seem to resemble anything.
How to approach this problem?