Answer
See explanation
Work Step by Step
$\begin{align*}
M_X(t) &= E(e^{Xt}) \\
&= \displaystyle \sum_{-\infty}^\infty e^{kt}P(X=k) \\
&= \displaystyle \sum_{k=0}^{n-1} e^{kt}p_X(k) & \text{[ since }\ p_X(k) = 0, k\not\in \{0,1,2,3,\dots,n-1\}\ ] \\
&= \displaystyle \sum_{k=0}^{n-1}e^{kt}\cdot\left(\frac{1}{n}\right) & \text{[ since }\ p_X(k) = \frac{1}{n}, k\in \{0,1,2,3,\dots,n-1\}\ ] \\
&= \displaystyle \left(\frac{1}{n}\right) \cdot \underbrace{\sum_{k=0}^{n-1} e^{kt}}_{S_n\ \text{of G.P.:}\ t_1\,=\,1,\ r\,=\,e^t} \\
&= \displaystyle \left(\frac{1}{n}\right) \cdot \overbrace{\frac{1-(e^t)^n}{1-e^t}}^{S_n = \frac{t_1(1-r^n)}{1-r}} \\
\color{blue}{M_X(t)}\ &\color{blue}{= \frac{1-e^{nt}}{n(1-e^t)}}
\end{align*}$
