expected value $E[X]$ from MGF. How to differentiate this 
$$\sum_{x=a}^{a+n-1}\frac1ne^{tx}$$
for $m(t)$ and find $E[X]$ from this?

Presumably, the expression given is the moment generating function of a discrete, uniform random variable. You can differentiate (with respect to $t$) term by term to get $m'(t) = \sum_{x=a}^{a+n-1} \frac{1}{n}xe^{tx}$. From here, set $t=0$. Then you are left with an arithmetic series to evaluate. 
