Answer
$-5,050$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given expression, $
\displaystyle\sum_{k=1}^{100} -k
,$ use the summation properties and then simplify.
$\bf{\text{Solution Details:}}$
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n ci=c\displaystyle\sum_{i=1}^n i,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\displaystyle\sum_{k=1}^{100} (-1)k
\\\\=
-1\displaystyle\sum_{k=1}^{100} k
\\\\=
-\displaystyle\sum_{k=1}^{100} k
.\end{array}
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n i=\dfrac{n(n+1)}{2},$ with $n=
100
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
-\dfrac{n(n+1)}{2}
\\\\=
-\dfrac{100(100+1)}{2}
\\\\=
-5,050
.\end{array}
