Let tilde$(x)[n]$ be a periodic sequence with fundamental period $N$ and let tilde$(x)[k]$ be its discrete Fourier series coefficient. Let widetilde$(x{)}_3[n]$ be periodic with period $3N$ consisting of three periods of tilde$(x)[n]$ and let widetilde$(x{)}_3[k]$ be its discrete Fourier series coefficient.

$($a$)$ Determine widetilde$(x{)}_3[k]$ in terms of widetilde$(x)[k].$

$($b$)$ Let tilde$(x)[n]=\{dots,1,3,1,3,1,3,1,3,$dots$\}.$ Verify your results in part $($a$)$ by explicitly computing tilde$(x)[k]$ and widetilde$(x{)}_3[k].$