Alright, let's tackle the summation problem step by step.
Given the expression:
[tex]\[
3 \sum_{k=1}^n (2k - 3) + \sum_{k=1}^n (4 - 5k)
\][/tex]
where [tex]\( n = 6 \)[/tex].
### Step 1: Calculate the First Summation Component
First, we consider the summation inside the first term:
[tex]\[
\sum_{k=1}^n (2k - 3)
\][/tex]
Substitute [tex]\( n = 6 \)[/tex]:
[tex]\[
\sum_{k=1}^6 (2k - 3)
\][/tex]
Calculate each term inside the summation for [tex]\( k = 1, 2, 3, 4, 5, 6 \)[/tex]:
- When [tex]\( k = 1 \)[/tex]: [tex]\( 2 \cdot 1 - 3 = -1 \)[/tex]
- When [tex]\( k = 2 \)[/tex]: [tex]\( 2 \cdot 2 - 3 = 1 \)[/tex]
- When [tex]\( k = 3 \)[/tex]: [tex]\( 2 \cdot 3 - 3 = 3 \)[/tex]
- When [tex]\( k = 4 \)[/tex]: [tex]\( 2 \cdot 4 - 3 = 5 \)[/tex]
- When [tex]\( k = 5 \)[/tex]: [tex]\( 2 \cdot 5 - 3 = 7 \)[/tex]
- When [tex]\( k = 6 \)[/tex]: [tex]\( 2 \cdot 6 - 3 = 9 \)[/tex]
Now add these values together:
[tex]\[
-1 + 1 + 3 + 5 + 7 + 9 = 24
\][/tex]
Thus, the first summation component is:
[tex]\[
\sum_{k=1}^6 (2k - 3) = 24
\][/tex]
Multiplying this by 3 gives:
[tex]\[
3 \sum_{k=1}^6 (2k - 3) = 3 \cdot 24 = 72
\][/tex]
### Step 2: Calculate the Second Summation Component
Next, we consider the second summation:
[tex]\[
\sum_{k=1}^n (4 - 5k)
\][/tex]
Substitute [tex]\( n = 6 \)[/tex]:
[tex]\[
\sum_{k=1}^6 (4 - 5k)
\][/tex]
Calculate each term inside the summation for [tex]\( k = 1, 2, 3, 4, 5, 6 \)[/tex]:
- When [tex]\( k = 1 \)[/tex]: [tex]\( 4 - 5 \cdot 1 = -1 \)[/tex]
- When [tex]\( k = 2 \)[/tex]: [tex]\( 4 - 5 \cdot 2 = -6 \)[/tex]
- When [tex]\( k = 3 \)[/tex]: [tex]\( 4 - 5 \cdot 3 = -11 \)[/tex]
- When [tex]\( k = 4 \)[/tex]: [tex]\( 4 - 5 \cdot 4 = -16 \)[/tex]
- When [tex]\( k = 5 \)[/tex]: [tex]\( 4 - 5 \cdot 5 = -21 \)[/tex]
- When [tex]\( k = 6 \)[/tex]: [tex]\( 4 - 5 \cdot 6 = -26 \)[/tex]
Now add these values together:
[tex]\[
-1 + (-6) + (-11) + (-16) + (-21) + (-26) = -81
\][/tex]
Thus, the second summation component is:
[tex]\[
\sum_{k=1}^6 (4 - 5k) = -81
\][/tex]
### Step 3: Combine Both Summation Components
Finally, we combine the results of both summations:
[tex]\[
3 \sum_{k=1}^6 (2k - 3) + \sum_{k=1}^6 (4 - 5k) = 72 + (-81) = -9
\][/tex]
### Final Answer
So, the value of the given summation when [tex]\( n = 6 \)[/tex] is:
[tex]\[
-9
\][/tex]
