Very simple question about signal periodicity
no, but $x[n]$ is, what we call, an "even-symmetry" function:

$$ x[-n] = x[n] \quad \forall n \in \mathbb{Z} \ . $$

what makes $x[n]$ periodic is $$ x[n+N] = x[n] \quad \forall n \in \mathbb{Z} $$ which is really the premise for the DFT of length $N$.