Question 13
Let the sequence of rationals $x_1,$ $x_2,$ $\ldots$ be defined such that $x_1 = \frac{25}{11}$ and $$
x_{k+1} = \frac{1}{3}\left(x_k + \frac{1}{x_k} - 1\right)
$$ for all $k \ge 1.$ Then $x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n.$ Find the remainder when $m + n$ is divided by $1000.$
x_{k+1} = \frac{1}{3}\left(x_k + \frac{1}{x_k} - 1\right)
$$ for all $k \ge 1.$ Then $x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n.$ Find the remainder when $m + n$ is divided by $1000.$