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.

Correct me if I'm wrong, but it looks like $a,b,c$ and $n$ are all constants inside that sum.  So
$$
S = \sum_{i=1}^n a^{\frac{i(c - b)}{n}} = \sum_{i=1}^n \left(a^{\frac{c - b}{n}}\right)^i
$$
which is just a finite geometric sum.  Letting $r = a^{\frac{c - b}{n}}$, this sums to $\frac{1 - r^{n+1}}{1 - r} - 1$.
