To find the sum of the series [tex]\(\sum_{k=1}^4 (-4k)\)[/tex], we will break it down step-by-step.
1. Identify the sequence terms:
The sum [tex]\(\sum_{k=1}^4 (-4k)\)[/tex] can be expanded to show each term individually:
[tex]\[
(-4 \cdot 1) + (-4 \cdot 2) + (-4 \cdot 3) + (-4 \cdot 4)
\][/tex]
2. Calculate each term:
- When [tex]\(k = 1\)[/tex]: [tex]\(-4 \cdot 1 = -4\)[/tex]
- When [tex]\(k = 2\)[/tex]: [tex]\(-4 \cdot 2 = -8\)[/tex]
- When [tex]\(k = 3\)[/tex]: [tex]\(-4 \cdot 3 = -12\)[/tex]
- When [tex]\(k = 4\)[/tex]: [tex]\(-4 \cdot 4 = -16\)[/tex]
3. Write out the expanded sum:
[tex]\[
-4 + (-8) + (-12) + (-16)
\][/tex]
4. Sum the individual terms:
[tex]\[
-4 - 8 - 12 - 16 = -40
\][/tex]
As a result, the sum of the series [tex]\(\sum_{k=1}^4 (-4k)\)[/tex] is [tex]\(-40\)[/tex].
The correct answer is [tex]\( \boxed{-40} \)[/tex].
