Certainly! Let's find the sum of the series [tex]\(\sum_{k=0}^4 (-2k)\)[/tex].
1. Identify the series and the range of summation:
We need to find the sum from [tex]\(k = 0\)[/tex] to [tex]\(k = 4\)[/tex] for the expression [tex]\(-2k\)[/tex]. This means we will calculate:
[tex]\[
(-2 \cdot 0) + (-2 \cdot 1) + (-2 \cdot 2) + (-2 \cdot 3) + (-2 \cdot 4)
\][/tex]
2. Calculate each term separately:
[tex]\[
\begin{aligned}
&-2 \cdot 0 = 0, \\
&-2 \cdot 1 = -2, \\
&-2 \cdot 2 = -4, \\
&-2 \cdot 3 = -6, \\
&-2 \cdot 4 = -8.
\end{aligned}
\][/tex]
3. Sum the calculated terms together:
[tex]\[
0 + (-2) + (-4) + (-6) + (-8).
\][/tex]
4. Perform the addition step-by-step:
[tex]\[
\begin{aligned}
&0 + (-2) = -2, \\
&-2 + (-4) = -6, \\
&-6 + (-6) = -12, \\
&-12 + (-8) = -20.
\end{aligned}
\][/tex]
5. Final sum:
[tex]\[
-20
\][/tex]
Therefore, the sum of the series [tex]\(\sum_{k=0}^4 (-2k)\)[/tex] is [tex]\(\boxed{-20}\)[/tex].
