To solve the problem of finding the sum [tex]\(\sum_{k=1}^6 (3 - k)\)[/tex], let's break it down step-by-step.
1. Identify the given expression for each term in the sum:
We are given the expression [tex]\((3 - k)\)[/tex] where [tex]\(k\)[/tex] takes on successive integer values starting from 1 up to 6.
2. Calculate each term individually for [tex]\(k\)[/tex] from 1 to 6:
- When [tex]\(k = 1\)[/tex]: [tex]\(3 - 1 = 2\)[/tex]
- When [tex]\(k = 2\)[/tex]: [tex]\(3 - 2 = 1\)[/tex]
- When [tex]\(k = 3\)[/tex]: [tex]\(3 - 3 = 0\)[/tex]
- When [tex]\(k = 4\)[/tex]: [tex]\(3 - 4 = -1\)[/tex]
- When [tex]\(k = 5\)[/tex]: [tex]\(3 - 5 = -2\)[/tex]
- When [tex]\(k = 6\)[/tex]: [tex]\(3 - 6 = -3\)[/tex]
3. Sum these values together:
[tex]\[
2 + 1 + 0 - 1 - 2 - 3
\][/tex]
4. Combine the terms:
[tex]\[
2 + 1 = 3
\][/tex]
[tex]\[
3 + 0 = 3
\][/tex]
[tex]\[
3 - 1 = 2
\][/tex]
[tex]\[
2 - 2 = 0
\][/tex]
[tex]\[
0 - 3 = -3
\][/tex]
So, the sum [tex]\(\sum_{k=1}^6 (3 - k)\)[/tex] is [tex]\(-3\)[/tex].
Among the given options, the correct answer is:
B) [tex]\(-3\)[/tex].
