Find Fourier coefficients of discrete odd signal

Question

I'm self studying signal and system. I've come across this problem: if $a_1 = 1, a_2 = j$, what are $a_3, a_4, a_5$ for a discrete odd signal x[n] with fundamental period of N=6?

Answer 1

Since the signal is odd we have $$a_n=-a_{-n}$$also $$a_n=a_{n+N}$$therefore $$a_5=a_{-1}=-a_1=-1\\a_4=a_{-2}=-a_2=-j$$and $$a_0=-a_0=0\\a_3=-a_{-3}=-a_{6-3}=-a_3\to a_3=0$$finally here are the coefficients $$(a_0,a_1,a_2,a_3,a_4,a_5)=(0,1,j,0,-j,-1)$$