sum of 17 terms

The common difference, $d=6\frac{3}{4}-5\frac{1}{2}=\frac{5}{4}$ .

The sum of the terms of an arithmetic progression is given by $s=\frac{n}{2}[2a_1+(n-1)(d)]$ where $a_1$ is the first term, $n$ is the number of terms and $d$ is the common difference. Substituting, we have

$s=\frac{17}{2}\left[2\left(5\frac{1}{2}\right)+16\left(\frac{5}{4}\right)\right]=\boxed{263\frac{1}{2}}$