The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$ (1)Find the general formula of $\{a_n\}$; (2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.
Source: 2022 China Southeast Grade 10 P1
Tags: algebra, Sequence, induction
The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$ (1)Find the general formula of $\{a_n\}$; (2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.