Answered

At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

Evaluate the following expression:

[tex]\[ \sum_{k=1}^4 k - \sum_{k=2}^8 (2k - 3) + \sum_{k=5}^{12} 2k^2 \][/tex]

Sagot :

GinnyAnswer
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.