$x_{n+1}=x_n+\sqrt{x_n^2-1} \to x_n=\frac{x_{n+1}^2+1}{2x_{n+1}}$ $ \frac{x_n+1}{x_n-1}=\frac{(x_{n+1}+1)^2}{(x_{n+1}-1)^2} \to \frac{x_1+1}{x_1-1}=(\frac{x_n+1}{x_n-1})^{2^{n-1}}$ So let $x_{2024}=a$ then $\frac{x_1+1}{x_1-1}=(\frac{a+1}{a-1})^{2^{2023}}=b \to x_1=\frac{b+1}{b-1}=\frac{(a+1)^{2^{2023}}+(a-1)^{2^{2023}}}{(a+1)^{2^{2023}}-(a-1)^{2^{2023}}}$ for some rational $a>1$