Who can elaborate on what the $+$ operation represents on $\{0, 1\}$? If $+$ is understood as in $\mathbb{F}_2$, then define $y_n = x_{2n} + x_{2n + 99} + x_{2n + 100}$, then it can be obtained that $y_2 = 1$ and $y_n = 0$ for $n > 100$, which indicates that $x$ is not periodic.

lbh_qys wrote: Who can elaborate on what the $+$ operation represents on $\{0, 1\}$? If $+$ is understood as in $\mathbb{F}_2$, then define $y_n = x_{2n} + x_{2n + 99} + x_{2n + 100}$, then it can be obtained that $y_2 = 1$ and $y_n = 0$ for $n > 100$, which indicates that $x$ is not periodic. In original formulation were words "The sequence of remainders by module $2$...". So I think that all operations here are by $\pmod 2$.