If $$i = \sqrt { - 1} ,$$  the number of values of $${i^n} + {i^{ - n}}$$  for different $$n \in Z$$  is

Solution :
If $$n=0$$  we get the answer as 2
If $$n$$ is even and $$n$$ is simply a multiple of 2 but not a multiple of 4, we get the answer as $$-$$ 2
If $$n$$ is odd,
Let $$n=3$$  we get
$$\eqalign{ & - i - \frac{1}{i} \cr & = \frac{{1 - 1}}{i} \cr & = 0 \cr & n = 5{\text{ we get}} \cr & i + \frac{1}{i} \cr & = 0 \cr} $$
Hence in total there are only 3 values, $$ - 2,\,0,\,2$$