Let $ x[n] $ be a real periodic sequence with fundamental period $ {N}_{0} $ and Fourier coefficients $ {c}_{k}={a}_{k}+j{b}_{k} $ where ak and bk are both real.
Show that $ {a}_{-k}={a}_{k} $ and $ {b}_{-k}=-{b}_{k} $.
If $ x[n] $ is real we have (equation for Fourier coefficients):
$ {c}_{-k}=\frac{1}{{N}_{0}}\sum_{n=0}^{{N}_{0}-1}x[n]{e}^{jk{\omega}_{0}n} $
and further:
$ ={\left(\frac{1}{{N}_{0}}\sum_{n=0}^{{N}_{0}-1}x[n]{e}^{-jk{\omega}_{0}n} \right)}^{*}={{c}^{*}}_{k} $
Therefore:
$ {c}_{-k}={a}_{-k}+j{b}_{-k}={({a}_{k}+{b}_{k})}^{*}={a}_{k}-j{b}_{k} $
So now we can see that:
</math> {a}_{-k}={a}_{k} and {b}_{-k}=-{b}_{k}</math>
