distribution of the mean of an exponential distribution (mgf)

Let $\displaystyle \overline{X}$ be the mean of a random sample from the exponential distribution, Exp($\displaystyle \theta$). Using the mgf technique determine the distribution of $\displaystyle \overline{X}$.

$\displaystyle M_{\overline{X}}(t) = Ee^{t\overline{X}}$

$\displaystyle M_{\overline{X}}(t) = Ee^{\frac{t}{n}\sum{X_i}}$

$\displaystyle M_{\overline{X}}(t) = e^{\frac{1}{n}}(1-\frac{t}{\theta})^{-n}$ I'm sure this is wrong, but I don't know why.

The book has the answer in the back, saying it should be $\displaystyle Gamma(\alpha = n, \beta = \frac{\theta}{n})$, but I don't know how to get there. MGF problems often seem to give me difficulty. :(