$x_k \in\mathbb{Q^+}, \ \forall k.$ $ \ $ For all $k \ge2$ we have $$x_{k+1}=\frac{x_k}{x_1^2+\cdots+x_k^2},$$$$x_{k}=\frac{x_{k-1}}{x_1^2+...+x_{k-1}^2}.$$$$\Rightarrow x_{k+1}=\frac{x_k}{x_k^2+\frac{x_{k-1}}{x_k}}= \frac{x_k^2}{x_k^3+x_{k-1}}.$$$$\Rightarrow x_k^4+4\frac{x_{k-1}}{x_{k+1}} = \left(\frac{x_k^3+2x_{k-1}}{x_k} \right)^2.$$