True/False Question :
Let $\left \{ a_{n} \right \}^{\infty }_{n=1}$ be a sequence of elements in $\left \{ 0,1 \right \}$ such that for all positive integers $n$, $\sum_{i=n}^{n+9}a_{i}$ is divisible by $3$. Then there exists a positive integer $k$ such that $a_{n+k}=a_{n}$ for all positive integers $n$.
