Answer :
Certainly! Let's break down the problem and solve each component of the sum step by step.
### Step 1: Calculate the Sum [tex]$\sum_{k=1}^4 k$[/tex]
This is the sum of the first four natural numbers:
[tex]\[ 1 + 2 + 3 + 4 \][/tex]
Adding these values together:
[tex]\[ 1 + 2 + 3 + 4 = 10 \][/tex]
So,
[tex]\[ \sum_{k=1}^4 k = 10 \][/tex]
### Step 2: Calculate the Sum [tex]$\sum_{k=2}^8 (2k - 3)$[/tex]
This is the sum from [tex]\(k = 2\)[/tex] to [tex]\(k = 8\)[/tex] of the expression [tex]\(2k - 3\)[/tex]:
[tex]\[ (2(2) - 3) + (2(3) - 3) + (2(4) - 3) + (2(5) - 3) + (2(6) - 3) + (2(7) - 3) + (2(8) - 3) \][/tex]
Plugging in the values and simplifying inside the sum:
[tex]\[ (4 - 3) + (6 - 3) + (8 - 3) + (10 - 3) + (12 - 3) + (14 - 3) + (16 - 3) \][/tex]
Simplify each term:
[tex]\[ 1 + 3 + 5 + 7 + 9 + 11 + 13 \][/tex]
Adding these results together:
[tex]\[ 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 \][/tex]
So,
[tex]\[ \sum_{k=2}^8 (2k - 3) = 49 \][/tex]
### Step 3: Calculate the Sum [tex]$\sum_{k=5}^{12} 2k^2$[/tex]
This is the sum from [tex]\(k = 5\)[/tex] to [tex]\(k = 12\)[/tex] of the expression [tex]\(2k^2\)[/tex]:
[tex]\[ 2(5^2) + 2(6^2) + 2(7^2) + 2(8^2) + 2(9^2) + 2(10^2) + 2(11^2) + 2(12^2) \][/tex]
Plugging in the values:
[tex]\[ 2(25) + 2(36) + 2(49) + 2(64) + 2(81) + 2(100) + 2(121) + 2(144) \][/tex]
Simplifying each term:
[tex]\[ 50 + 72 + 98 + 128 + 162 + 200 + 242 + 288 \][/tex]
Adding these results together:
[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 need to evaluate the expression combining these sums:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 \][/tex]
Substitute the sums we calculated:
[tex]\[ 10 - 49 + 1240 \][/tex]
Calculate the final result:
[tex]\[ 10 - 49 = -39 \][/tex]
[tex]\[ -39 + 1240 = 1201 \][/tex]
Thus, the final result is:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 = 1201 \][/tex]
So, the individual sums are:
[tex]\[ \sum_{k=1}^4 k = 10 \][/tex]
[tex]\[ \sum_{k=2}^8 (2k - 3) = 49 \][/tex]
[tex]\[ \sum_{k=5}^{12} 2k^2 = 1240 \][/tex]
And the combined result is:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 = 1201 \][/tex]
### Step 1: Calculate the Sum [tex]$\sum_{k=1}^4 k$[/tex]
This is the sum of the first four natural numbers:
[tex]\[ 1 + 2 + 3 + 4 \][/tex]
Adding these values together:
[tex]\[ 1 + 2 + 3 + 4 = 10 \][/tex]
So,
[tex]\[ \sum_{k=1}^4 k = 10 \][/tex]
### Step 2: Calculate the Sum [tex]$\sum_{k=2}^8 (2k - 3)$[/tex]
This is the sum from [tex]\(k = 2\)[/tex] to [tex]\(k = 8\)[/tex] of the expression [tex]\(2k - 3\)[/tex]:
[tex]\[ (2(2) - 3) + (2(3) - 3) + (2(4) - 3) + (2(5) - 3) + (2(6) - 3) + (2(7) - 3) + (2(8) - 3) \][/tex]
Plugging in the values and simplifying inside the sum:
[tex]\[ (4 - 3) + (6 - 3) + (8 - 3) + (10 - 3) + (12 - 3) + (14 - 3) + (16 - 3) \][/tex]
Simplify each term:
[tex]\[ 1 + 3 + 5 + 7 + 9 + 11 + 13 \][/tex]
Adding these results together:
[tex]\[ 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 \][/tex]
So,
[tex]\[ \sum_{k=2}^8 (2k - 3) = 49 \][/tex]
### Step 3: Calculate the Sum [tex]$\sum_{k=5}^{12} 2k^2$[/tex]
This is the sum from [tex]\(k = 5\)[/tex] to [tex]\(k = 12\)[/tex] of the expression [tex]\(2k^2\)[/tex]:
[tex]\[ 2(5^2) + 2(6^2) + 2(7^2) + 2(8^2) + 2(9^2) + 2(10^2) + 2(11^2) + 2(12^2) \][/tex]
Plugging in the values:
[tex]\[ 2(25) + 2(36) + 2(49) + 2(64) + 2(81) + 2(100) + 2(121) + 2(144) \][/tex]
Simplifying each term:
[tex]\[ 50 + 72 + 98 + 128 + 162 + 200 + 242 + 288 \][/tex]
Adding these results together:
[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 need to evaluate the expression combining these sums:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 \][/tex]
Substitute the sums we calculated:
[tex]\[ 10 - 49 + 1240 \][/tex]
Calculate the final result:
[tex]\[ 10 - 49 = -39 \][/tex]
[tex]\[ -39 + 1240 = 1201 \][/tex]
Thus, the final result is:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 = 1201 \][/tex]
So, the individual sums are:
[tex]\[ \sum_{k=1}^4 k = 10 \][/tex]
[tex]\[ \sum_{k=2}^8 (2k - 3) = 49 \][/tex]
[tex]\[ \sum_{k=5}^{12} 2k^2 = 1240 \][/tex]
And the combined result is:
[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 = 1201 \][/tex]
