TUYMAADA 2024 S3

All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$ -th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$ .) Is there an integer $n$ such that all the numbers $$f(n),f(2n),f(3n),\dots,f(10n^2)$$ are perfect squares?