Question 53
Consider the sequence $a_{1}, a_{2}, a_{3}, \ldots$ defined by $a_{1} = 2024^{2024}$ and for each positive integer $n$,
\[
a_{n + 1} = \left|a_{n} - \sqrt{2}\right|.
\]
Prove that there exists an integer $k$ such that $a_{k + 2} = a_{k}$.
Here $|x|$ denotes the absolute value of $x$.
\[
a_{n + 1} = \left|a_{n} - \sqrt{2}\right|.
\]
Prove that there exists an integer $k$ such that $a_{k + 2} = a_{k}$.
Here $|x|$ denotes the absolute value of $x$.