Question 46
Determine all sequences $(a_n)_{n \ge 1}$ of positive rational numbers satisfying
$$
a_{k+1} = \frac{a_k^2 + a_{k+2}^2}{a_k + a_{k+2}} \quad \text{for all } k \ge 1. \quad (*)
$$
$$
a_{k+1} = \frac{a_k^2 + a_{k+2}^2}{a_k + a_{k+2}} \quad \text{for all } k \ge 1. \quad (*)
$$