How to evaluate the following summation I am trying to find the definite integral of $a^x$ between $b$ and $c$ as the limit of a Riemann sum (where $a > 0$):
$I = \displaystyle\int_b^c \! a^{x} \, \mathrm{d}x.$
However, I'm currently stuck in the following part, in order to find S:
$S = \displaystyle\sum\limits_{i=1}^n \displaystyle{a^{\displaystyle\frac{i(c-b)}{n}}}$
Is there a formula for this kind of expression? Thank you for your help.

Note that: 
$$\begin{align}
S &= \sum\limits_{i=1}^n \displaystyle{a^{\displaystyle\frac{i(c-b)}{n}}} \\
&= \sum\limits_{i=1}^n \left(a^{\left(\dfrac{(c-b)}{n}\right)}\right)^i
\end{align}$$
Now, we can use the finite form of the geometric series formula:
$$S = \frac{\left(a^{\left(\dfrac{(c-b)}{n}\right)}\right)-\left(a^{\left(\dfrac{(c-b)}{n}\right)}\right)^{n+1}}{1-a^{\left(\dfrac{(c-b)}{n}\right)}}$$
...and that limit will be pretty nasty, but do-able.  (I think.)
