Simple proof for Fourier transformation (DFT)

Question

The question is in the image below: Image

Answer 1

$$y_k = \sum_{n=0}^{N-1} x_n \exp(-2 \pi i kn /N)$$

Notice that $\exp(0) = 1$, by setting $k=0$ in the above equation,

\begin{align}y_0 &= \sum_{n=0}^{N-1} x_n \exp(-2 \pi i 0n /N)\\&=\sum_{n=0}^{N-1}x_n \exp(0)\\&=\sum_{n=0}^{N-1} x_n \end{align}