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.

This might help.  Suppose $y = a^x$. Then $\ln(y) = x\ln(a)$. Differentiating you get
$${y'\over y} = \ln(a),$$
so $$ y' = y\ln(a) = a^x \ln(a).$$
Hence 
$$\int a^x\, dx = {a^x\over \ln(a)} + C.$$
