Pure imaginary numbers : $ ( j^{2n} - j^n ) \in i\Bbb R $? Considering the complex number $j$ such that $$ j = \frac{-1}{2} + i\frac{\sqrt3}{2} $$
Prove that $ \forall n \in \Bbb Z : $
$$ ( j^{2n} - j^n ) \in i\Bbb R $$
( $i\Bbb R$ being the set of pure imaginary numbers)


Hint:
Note that using Euler's formula, we can write $$ j = \cos \frac{2\pi}3 + i\sin \frac{2\pi}3 = e^{\dfrac{2\pi i} 3}$$ which is a complex root of unity. 
