# An eventually periodic sequence

**Number Theory**Level pending

Let us call a real sequence \(\{a_n\}_{n\ge 0}\) **eventually periodic** if \(\exists \) positive numbers \(m,N\) such that \(a_{n+N}=a_n,\ \forall n\ge m\). The smallest such \(N\) is called the **period** of the eventually periodic sequence.

Is the following sequence eventually periodic? \[ c_0=\frac{1}{2017},\quad c_{n}=|1-|1-2c_{n-1}||,\quad n\ge 1 \]

If it is, find its period.

Put \(666\) as your answer if you think that the sequence is not eventually periodic.