Answer
$S_{20}=-\dfrac{75}{2}$
Work Step by Step
Using $S_{n}=\dfrac{n}{2}(2a_1+(n-1)d)$, the sum of the first $
20
$ terms of the sequence $
\dfrac{1}{2},\dfrac{1}{4},0,...,-\dfrac{17}{4}
$ is
\begin{array}{l}
S_{20}=\dfrac{20}{2}\left(2\cdot\dfrac{1}{2}+(20-1)\left(-\dfrac{1}{4}\right)\right)\\\\
S_{20}=10\left(1-\dfrac{19}{4}\right)\\\\
S_{20}=-\dfrac{75}{2}
.\end{array}