The sequence $\{a_n\}_{n\ge 1}$ of real numbers is defined as follows: $$a_1=1, \quad \text{and}\quad a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \quad \text{for all} \quad n\ge 1$$Find $a_{2024}$.
Source: Philippine Mathematical Olympiad 2024 P6
Tags: floor function, number theory
The sequence $\{a_n\}_{n\ge 1}$ of real numbers is defined as follows: $$a_1=1, \quad \text{and}\quad a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \quad \text{for all} \quad n\ge 1$$Find $a_{2024}$.