Periodic
$ x[n]= $$ j^{n} $ is a discrete time (DT) periodic signal. It's period is 4*k, where k is an integer. However, it's fundamental period is 4.
$ j^{1}=-1 $
$ j^{2}=-j $
$ j^{3}=1 $
$ j^{4}=j $
$ j^{5}=-1 $
$ j^{6}=-j $
$ j^{7}=1 $
$ j^{8}=j $
Non-Periodic
$ x[n]=\cos{n} $ is an example of a non-periodoc signal because there is not integer value for n such that $ x[n+N]=x[n] $. It would be periodic if $ N=K*2\pi $, but $ \pi $ is not an integer and therefore can not be chosen.
