The definition of a periodic DT signal is that there exists an integer N such that $ x[n+N] = x[n] $ for all $ n $.
For example, $ x[n] = j^n $ is periodic. One can show that:
$ x[1] = j\, $
$ x[2] = -1\, $
$ x[3] = -j\, $
$ x[4] = 1\, $
$ x[5] = j\, $
$ x[6] = -1\, $
$ x[7] = -j\, $
$ x[8] = 1\, $
