To solve the given summation [tex]\(\sum_{n=2}^{12}(8-2n)\)[/tex], let's break it down step-by-step.
1. Identify the series terms:
We need to evaluate the expression [tex]\(8 - 2n\)[/tex] for each integer value of [tex]\(n\)[/tex] from [tex]\(2\)[/tex] to [tex]\(12\)[/tex].
2. Compute individual terms:
Let's find the terms in the summation sequence:
- For [tex]\(n = 2\)[/tex]: [tex]\(8 - 2(2) = 8 - 4 = 4\)[/tex]
- For [tex]\(n = 3\)[/tex]: [tex]\(8 - 2(3) = 8 - 6 = 2\)[/tex]
- For [tex]\(n = 4\)[/tex]: [tex]\(8 - 2(4) = 8 - 8 = 0\)[/tex]
- For [tex]\(n = 5\)[/tex]: [tex]\(8 - 2(5) = 8 - 10 = -2\)[/tex]
- For [tex]\(n = 6\)[/tex]: [tex]\(8 - 2(6) = 8 - 12 = -4\)[/tex]
- For [tex]\(n = 7\)[/tex]: [tex]\(8 - 2(7) = 8 - 14 = -6\)[/tex]
- For [tex]\(n = 8\)[/tex]: [tex]\(8 - 2(8) = 8 - 16 = -8\)[/tex]
- For [tex]\(n = 9\)[/tex]: [tex]\(8 - 2(9) = 8 - 18 = -10\)[/tex]
- For [tex]\(n = 10\)[/tex]: [tex]\(8 - 2(10) = 8 - 20 = -12\)[/tex]
- For [tex]\(n = 11\)[/tex]: [tex]\(8 - 2(11) = 8 - 22 = -14\)[/tex]
- For [tex]\(n = 12\)[/tex]: [tex]\(8 - 2(12) = 8 - 24 = -16\)[/tex]
So the terms are: [tex]\([4, 2, 0, -2, -4, -6, -8, -10, -12, -14, -16]\)[/tex].
3. Sum the series:
Next, we sum the computed terms:
[tex]\[
4 + 2 + 0 + (-2) + (-4) + (-6) + (-8) + (-10) + (-12) + (-14) + (-16)
\][/tex]
Let's group and add these terms step-by-step for clarity:
[tex]\[
(4 + 2 + 0 + (-2)) + (-4 + (-6)) + (-8 + (-10)) + (-12 + (-14)) + (-16)
\][/tex]
Simplifying within each grouping:
[tex]\[
4 + 2 + 0 - 2 = 4
\][/tex]
[tex]\[
-4 + (-6) = -10
\][/tex]
[tex]\[
-8 + (-10) = -18
\][/tex]
[tex]\[
-12 + (-14) = -26
\][/tex]
[tex]\[
(-16)
\][/tex]
Summing these intermediate results:
[tex]\[
4 + (-10) + (-18) + (-26) + (-16)
\][/tex]
Further simplification step-by-step:
[tex]\[
4 - 10 = -6
\][/tex]
[tex]\[
-6 - 18 = -24
\][/tex]
[tex]\[
-24 - 26 = -50
\][/tex]
[tex]\[
-50 - 16 = -66
\][/tex]
4. Answer:
The sum [tex]\(\sum_{n=2}^{12}(8-2n)\)[/tex] evaluates to [tex]\(-66\)[/tex].
