Let ${a_n}$ be the sequence of rational numbers defined by $a_0 = 2021,$ and $a_{n+1}=a_{n}+\frac{2}{a_{n}},$ for all $n \geq 0.$ Can any $a_{n}, n \geq 1,$ be a square of a rational number?
Source: Hong Kong TST - HKTST 2022 1.1
Tags: recurrence relation, number theory, algebra
Let ${a_n}$ be the sequence of rational numbers defined by $a_0 = 2021,$ and $a_{n+1}=a_{n}+\frac{2}{a_{n}},$ for all $n \geq 0.$ Can any $a_{n}, n \geq 1,$ be a square of a rational number?