Answer :
Sure, let's solve the given expression step-by-step.
The expression we need to evaluate is:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 \][/tex]
### Step 1: Evaluate [tex]\(\sum_{k=1}^4 k\)[/tex]
This is the sum of the first four positive integers:
[tex]\[ 1 + 2 + 3 + 4 = 10 \][/tex]
So,
[tex]\[ \sum_{k=1}^4 k = 10 \][/tex]
### Step 2: Evaluate [tex]\(\sum_{k=2}^8 (2k - 3)\)[/tex]
We need to evaluate the summation of the expression [tex]\((2k - 3)\)[/tex] from [tex]\(k = 2\)[/tex] to [tex]\(k = 8\)[/tex].
Calculating each term:
[tex]\[ \begin{align*} k = 2: & \quad 2(2) - 3 = 4 - 3 = 1 \\ k = 3: & \quad 2(3) - 3 = 6 - 3 = 3 \\ k = 4: & \quad 2(4) - 3 = 8 - 3 = 5 \\ k = 5: & \quad 2(5) - 3 = 10 - 3 = 7 \\ k = 6: & \quad 2(6) - 3 = 12 - 3 = 9 \\ k = 7: & \quad 2(7) - 3 = 14 - 3 = 11 \\ k = 8: & \quad 2(8) - 3 = 16 - 3 = 13 \\ \end{align*} \][/tex]
Summing these values:
[tex]\[ 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 \][/tex]
So,
[tex]\[ \sum_{k=2}^8 (2k - 3) = 49 \][/tex]
### Step 3: Evaluate [tex]\(\sum_{k=5}^{12} 2k^2\)[/tex]
We need to evaluate the summation of the expression [tex]\(2k^2\)[/tex] from [tex]\(k = 5\)[/tex] to [tex]\(k = 12\)[/tex].
Calculating each term:
[tex]\[ \begin{align*} k = 5: & \quad 2(5^2) = 2(25) = 50 \\ k = 6: & \quad 2(6^2) = 2(36) = 72 \\ k = 7: & \quad 2(7^2) = 2(49) = 98 \\ k = 8: & \quad 2(8^2) = 2(64) = 128 \\ k = 9: & \quad 2(9^2) = 2(81) = 162 \\ k = 10: & \quad 2(10^2) = 2(100) = 200 \\ k = 11: & \quad 2(11^2) = 2(121) = 242 \\ k = 12: & \quad 2(12^2) = 2(144) = 288 \\ \end{align*} \][/tex]
Summing these values:
[tex]\[ 50 + 72 + 98 + 128 + 162 + 200 + 242 + 288 = 1240 \][/tex]
So,
[tex]\[ \sum_{k=5}^{12} 2k^2 = 1240 \][/tex]
### Step 4: Combine the results
Now, we can combine all the results to find the final value of the expression:
[tex]\[ 10 - 49 + 1240 \][/tex]
Calculating:
[tex]\[ 10 - 49 = -39 \][/tex]
[tex]\[ -39 + 1240 = 1201 \][/tex]
Thus, the final result is:
[tex]\[ \boxed{1201} \][/tex]
The expression we need to evaluate is:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 \][/tex]
### Step 1: Evaluate [tex]\(\sum_{k=1}^4 k\)[/tex]
This is the sum of the first four positive integers:
[tex]\[ 1 + 2 + 3 + 4 = 10 \][/tex]
So,
[tex]\[ \sum_{k=1}^4 k = 10 \][/tex]
### Step 2: Evaluate [tex]\(\sum_{k=2}^8 (2k - 3)\)[/tex]
We need to evaluate the summation of the expression [tex]\((2k - 3)\)[/tex] from [tex]\(k = 2\)[/tex] to [tex]\(k = 8\)[/tex].
Calculating each term:
[tex]\[ \begin{align*} k = 2: & \quad 2(2) - 3 = 4 - 3 = 1 \\ k = 3: & \quad 2(3) - 3 = 6 - 3 = 3 \\ k = 4: & \quad 2(4) - 3 = 8 - 3 = 5 \\ k = 5: & \quad 2(5) - 3 = 10 - 3 = 7 \\ k = 6: & \quad 2(6) - 3 = 12 - 3 = 9 \\ k = 7: & \quad 2(7) - 3 = 14 - 3 = 11 \\ k = 8: & \quad 2(8) - 3 = 16 - 3 = 13 \\ \end{align*} \][/tex]
Summing these values:
[tex]\[ 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 \][/tex]
So,
[tex]\[ \sum_{k=2}^8 (2k - 3) = 49 \][/tex]
### Step 3: Evaluate [tex]\(\sum_{k=5}^{12} 2k^2\)[/tex]
We need to evaluate the summation of the expression [tex]\(2k^2\)[/tex] from [tex]\(k = 5\)[/tex] to [tex]\(k = 12\)[/tex].
Calculating each term:
[tex]\[ \begin{align*} k = 5: & \quad 2(5^2) = 2(25) = 50 \\ k = 6: & \quad 2(6^2) = 2(36) = 72 \\ k = 7: & \quad 2(7^2) = 2(49) = 98 \\ k = 8: & \quad 2(8^2) = 2(64) = 128 \\ k = 9: & \quad 2(9^2) = 2(81) = 162 \\ k = 10: & \quad 2(10^2) = 2(100) = 200 \\ k = 11: & \quad 2(11^2) = 2(121) = 242 \\ k = 12: & \quad 2(12^2) = 2(144) = 288 \\ \end{align*} \][/tex]
Summing these values:
[tex]\[ 50 + 72 + 98 + 128 + 162 + 200 + 242 + 288 = 1240 \][/tex]
So,
[tex]\[ \sum_{k=5}^{12} 2k^2 = 1240 \][/tex]
### Step 4: Combine the results
Now, we can combine all the results to find the final value of the expression:
[tex]\[ 10 - 49 + 1240 \][/tex]
Calculating:
[tex]\[ 10 - 49 = -39 \][/tex]
[tex]\[ -39 + 1240 = 1201 \][/tex]
Thus, the final result is:
[tex]\[ \boxed{1201} \][/tex]
