Evaluate the following summation: ∑ t = 2 6 5 t + 4 t \sum_{t=2}^65t+\frac{4}{t}

So in this problem, we're asked to evaluate the following summation. And to do this, I'm going to start by thinking about what some of my rules are when it comes to dealing with sums. Because one of the things that I notice here is we have this five t plus four over t, and we have the sum going from t equals two to six. Now right off the bat, what I can do using the addition rule here is split this into two separate sums. So we're going to have the sum from t equals two to six of this first piece, five t. Then I'm going to add this from the sum of t equals to the six of the second piece here, which is four over t. So notice how I split this into two separate sums. And what I can do is take any constants that I have like this five and this four, and I can pull them outside of the sum. So doing this, we're going to have five on the outside multiplied by the sum from t equals two to six, then that's gonna be for t. Then we'll have plus, and I'll take this four and I'll pull up on the outside. So we'll have four times the sum from t equals two to six of one over t. So, you know, four times one will give us four over t again, but I'm pulling that to the outside. So now we can deal with these sum separately. Now I'm first gonna deal with the sum right here, which is the sum from t equals two to six. I'm going to do that for t, so we're just gonna see what we get. Now I'm doing this separately over here, so that way we can we don't have to deal with the mess of numbers over there. So doing this, we're going to go from two to six. So we have t here, so it's gonna be two plus three plus four plus five plus six. Now I can do this math here. So So we have two plus three, which is five. So we're going to have five plus four plus five plus six. And we have five plus five, which is 10 plus four. That's gonna give us 14 plus six, which is going to give us 20. So this is what happens when we do this sum, meaning this entire thing, it can be just written as 20. So we'll have five times 20 right over here. Now what I'm going to do is add this to the sum we have right there. And doing that, I'm going to evaluate this sum separately as well. So we have the sum from t equals two to six, and that's gonna be of one over t. So doing this, we're going to have one over two, plus one over three, plus one over four, plus one over five, plus one over six. So this right here is gonna be the sum when we do it. Now evaluating this sum, it would take us a while to get all, like, fractions or the denominators to be all the same. You could multiply out all these fractions and do that real quick. But just to go ahead and skip that step to make this a little bit faster, the result you should get is 29 over 20. If you went ahead and put these into your calculator, you would also get 1.45 as a decimal, but we're gonna go with 29 over 20 to keep this as a fraction. So 29 over 20 is the result that we get. So this sum is gonna be four multiplied by 29 over 20. And then all you'd wanna do is multiply and add these numbers together to get our final result. So if you went ahead and did this all here, if you went ahead and evaluated this, you have five times 20 plus four times 29 over 20, and that whole thing would come out to 529 over five as your fraction. So this would be the result and the solution to the problem. And that is how you can evaluate a sum like this. So the main thing to notice is how we were able to take this sum that initially consisted of these two pieces in one function and split them into two separate ones. We could pull out the constants and notice how that made dealing with these individual sums much more straightforward. So hope you found this video helpful, and let's move on and get some more practice.

8. Definite Integrals

Riemann Sums

Calculus

8. Definite Integrals

Riemann Sums

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Evaluate the following summation:
$\sum_{t = 2}^{6} 5 t + \frac{4}{t}$

$471 over 5$

$529 over 5$

$26 over 3$

$22 over 3$

Verified step by step guidance

First, identify the expression inside the summation: $ 6t + \frac{4}{t} $. This is the function you will be summing from $ t = 2 $ to $ t = 6 $.

Next, calculate the value of the expression for each integer value of $ t $ from 2 to 6. For example, when $ t = 2 $, substitute $ t $ into the expression to get $ 6(2) + \frac{4}{2} $.

Continue this process for $ t = 3, 4, 5, $ and $ 6 $, substituting each value into the expression $ 6t + \frac{4}{t} $ and calculating the result.

After calculating the expression for each value of $ t $, sum all these results together to find the total value of the summation.

Finally, simplify the sum if necessary to match one of the given answer choices, ensuring that the final expression is in the form $ \frac{a}{b} $ where $ a $ and $ b $ are integers.