To solve the given summation problem when [tex]\( n = 6 \)[/tex], we need to evaluate two separate summations and then combine them according to the expression given. Specifically, we need to compute:
[tex]\[ 3 \sum_{k=1}^n (2k-3) + \sum_{k=1}^n (4-5k) \][/tex]
Let's break this down into two parts and evaluate them step-by-step.
### Step 1: Evaluate [tex]\(\sum_{k=1}^n (2k-3)\)[/tex]
We need to find the sum of the sequence [tex]\(2k - 3\)[/tex] from [tex]\( k = 1 \)[/tex] to [tex]\( k = 6 \)[/tex]:
[tex]\[
\sum_{k=1}^6 (2k-3) = (2 \cdot 1 - 3) + (2 \cdot 2 - 3) + (2 \cdot 3 - 3) + (2 \cdot 4 - 3) + (2 \cdot 5 - 3) + (2 \cdot 6 - 3)
\][/tex]
Calculating each term individually:
[tex]\[
= (2 \cdot 1 - 3) + (2 \cdot 2 - 3) + (2 \cdot 3 - 3) + (2 \cdot 4 - 3) + (2 \cdot 5 - 3) + (2 \cdot 6 - 3)
\][/tex]
[tex]\[
= (2 - 3) + (4 - 3) + (6 - 3) + (8 - 3) + (10 - 3) + (12 - 3)
\][/tex]
[tex]\[
= -1 + 1 + 3 + 5 + 7 + 9
\][/tex]
Adding these results:
[tex]\[
-1 + 1 = 0
\][/tex]
[tex]\[
0 + 3 = 3
\][/tex]
[tex]\[
3 + 5 = 8
\][/tex]
[tex]\[
8 + 7 = 15
\][/tex]
[tex]\[
15 + 9 = 24
\][/tex]
Therefore, [tex]\(\sum_{k=1}^6 (2k-3) = 24\)[/tex].
### Step 2: Evaluate [tex]\(\sum_{k=1}^n (4-5k)\)[/tex]
Next, we need to find the sum of the sequence [tex]\(4 - 5k\)[/tex] from [tex]\( k = 1 \)[/tex] to [tex]\( k = 6 \)[/tex]:
[tex]\[
\sum_{k=1}^6 (4-5k) = (4 - 5 \cdot 1) + (4 - 5 \cdot 2) + (4 - 5 \cdot 3) + (4 - 5 \cdot 4) + (4 - 5 \cdot 5) + (4 - 5 \cdot 6)
\][/tex]
Calculating each term individually:
[tex]\[
= (4 - 5 \cdot 1) + (4 - 5 \cdot 2) + (4 - 5 \cdot 3) + (4 - 5 \cdot 4) + (4 - 5 \cdot 5) + (4 - 5 \cdot 6)
\][/tex]
[tex]\[
= (4 - 5) + (4 - 10) + (4 - 15) + (4 - 20) + (4 - 25) + (4 - 30)
\][/tex]
[tex]\[
= -1 - 6 - 11 - 16 - 21 - 26
\][/tex]
Adding these results:
[tex]\[
-1 - 6 = -7
\][/tex]
[tex]\[
-7 - 11 = -18
\][/tex]
[tex]\[
-18 - 16 = -34
\][/tex]
[tex]\[
-34 - 21 = -55
\][/tex]
[tex]\[
-55 - 26 = -81
\][/tex]
Therefore, [tex]\(\sum_{k=1}^6 (4-5k) = -81\)[/tex].
### Step 3: Combine the Results
Now we combine the results with the given constants:
[tex]\[ 3 \cdot \sum_{k=1}^6 (2k-3) + \sum_{k=1}^6 (4-5k) \][/tex]
Substituting the results we obtained:
[tex]\[
3 \cdot 24 + (-81)
\][/tex]
[tex]\[
72 - 81
\][/tex]
[tex]\[
= -9
\][/tex]
### Final Answer
Therefore, the value of the given summation when [tex]\( n = 6 \)[/tex] is:
[tex]\[
-9
\][/tex]
