Evaluate the following summation: ∑ k = 1 4 ( k 2 ) 2 \sum_{k=1}^4\left(\frac{k}{_{}2}\right)^2 ∑ k = 1 4 ( 2 k ) 2

in this problem we are given this summation right here, and we are asked to evaluate the following sum. Now to do this, we are going to take a look at these values that we have in the lower and upper index, so we can see that we go from k equals one to zero of this equation at k over two squared. So to do this, well, we're gonna start from one, and we're gonna increment all the way up to four and add each result. That's how a summation works. So we'll start with a k value of one, which is gonna give us one over two squared. Then we're going to add our next k up value, which would be two. So we go one, two, three, four. So we're gonna go two over two. That's gonna be squared. Then we're going to go and then this is what? Three over two squared. Then we're going to finish off at four, which is going to give us four over two squared. Now what I can do is square each one of these results. So we're going to have one squared over two squared, which comes out to one over four. And here, we're going to have two squared over two squared, which comes out to four over four. And you could reduce this to just be one, but I'm gonna keep this in the denominator here to make things a little bit easier for us. So then we're going to have plus, and then we'll have three squared three over two squared. Three squared over two squared, which comes up to nine over four. And then lastly, we're gonna have four squared over two squared, which comes up to 16 over four. Now what you can do at this point is add all of these numbers together. You can add one plus four plus nine plus 16, and you should get 30. So you're gonna get 30 over four, which all reduces down to 15 over two. So this right here would be our result and the solution to this problem. That is how you can solve this sum. Let's move on.

8. Definite Integrals

Riemann Sums

Business Calculus

8. Definite Integrals

Riemann Sums

Struggling with Business Calculus?

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Multiple Choice

Evaluate the following summation:
$\sum_{k=1}^4\left(\frac{k}{_{}2}\right)^2$

$\frac{15}{2}$

$15$

$\frac{17}{2}$

$\frac{17}{4}$

Verified step by step guidance

Step 1: Understand the summation notation. The given summation is $ \sum_{k=1}^4 \left( \frac{k}{2} \right)^2 $. This means you need to evaluate the expression $ \left( \frac{k}{2} \right)^2 $ for each integer value of $ k $ from 1 to 4, and then sum up the results.

Step 2: Break down the expression $ \left( \frac{k}{2} \right)^2 $. For each value of $ k $, divide $ k $ by 2, then square the result. For example, when $ k = 1 $, the expression becomes $ \left( \frac{1}{2} \right)^2 $.

Step 3: Compute the individual terms of the summation. Substitute $ k = 1, 2, 3, 4 $ into the expression $ \left( \frac{k}{2} \right)^2 $ to get the terms: $ \left( \frac{1}{2} \right)^2, \left( \frac{2}{2} \right)^2, \left( \frac{3}{2} \right)^2, \left( \frac{4}{2} \right)^2 $.

Step 4: Simplify each term. For example, $ \left( \frac{1}{2} \right)^2 = \frac{1}{4} $, $ \left( \frac{2}{2} \right)^2 = 1 $, and so on. Write down the simplified terms.

Step 5: Add the simplified terms together. Sum up the results from Step 4 to get the final value of the summation.