Is it possible for sum of three sequences $x_n + y_n+z_n$ to equal $0$?

Question

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?

Answer 1

Step 1: If $x_n = \tan(\phi)$ for some $\phi $ then $$ x_{n+1} = -\frac{2 \tan(\phi)}{1- \tan^2(\phi)} = - \tan(2 \phi) = \tan(-2\phi) \, . $$ Therefore, if we define $$ \alpha = \arctan(x_1), \, \beta = \arctan(y_1), \, \gamma = \arctan(z_1) \, , $$ then $$ x_n = \tan((-2)^{n-1} \alpha), \, y_n = \tan((-2)^{n-1} \beta), \, z_n = \tan((-2)^{n-1} \gamma) $$ for all $n$, as long as the sequences are defined, i.e. as long as no denominator in the sequence definitions becomes zero.

Step 2: Generally, $$ \tan (u) + \tan (v) + \tan (w) = \tan (u) \tan(v) \tan(w) $$ holds if and only if $u+v+w$ is an integral multiple of $\pi$, see for example Prove the identity if x+y+z=xyz on AoPS.

Step 3: The initial values $x_1 = 2$, $y_1 = 4$, $z_1 = 6/7$ satisfy $x_1 + y_1 + z_1 = x_1 y_1 z_1$. It follows that $\alpha + \beta + \gamma$ is an integral multiple of $\pi$. But then also $$ (-2)^{n-1} \alpha + (-2)^{n-1} \beta + (-2)^{n-1} \gamma = k \pi \text{ for some }k \in \Bbb Z $$ which in turn implies that $$ x_n + y_n + z_n = x_n y_n z_n $$ as long as the sequences are defined. It follows directly from the sequence definitions that the right-hand side is never zero, therefore $x_n + y_n + z_n$ can not be zero either.

Remark: One can also verify directly from the recursive definition that $$ x_{n+1} + y_{n+1} + z_{n+1} - x_{n+1} y_{n+1} z_{n+1} = \frac{2(1-x_n y_n - x_n z_n - y_n z_n)(x_n + y_n +z_n -x_ny_n z_n)}{(x_n^2-1)(y_n^2-1)(z_n^2-1)} $$ so that $$ x_n + y_n +z_n = x_ny_n z_n \implies x_{n+1} + y_{n+1} + z_{n+1} = x_{n+1} y_{n+1} z_{n+1} \, . $$