Evaluate the following summation: ∑ i = 1 4 i 2 − 3 i + 8 \sum_{i=1}^4i^2-3i+8 ∑ i = 1 4 i 2 − 3 i + 8

So in this problem, we're asked to evaluate the following summation. And doing this should be relatively straightforward. We're just taking this entire function that we have in here, this equation, and we're adding it to itself with different numbers plugged in. And we're starting from one, and then we're gonna go all the way up to four. So let's just go ahead and do that. Now I is the variable that we're looking at in this equation. So to do this summation, well, first off, we're going to take these i's and replace them with one. So it's going to give us one squared minus three times one plus eight. And we need to increment this all the way up to four. This is a summation, so we're adding it to the next piece. So we're going to have, and then that's gonna be one. In this case, it's gonna be two squared, and it's gonna be minus three times two plus eight. And then we're going to increment up again. So we're going to add, and it's going to be three squared, and it's gonna be minus three times three plus eight. And then we're going to finally go up to our last increment here, which stops at four. So we're going to go to four squared. It's gonna be minus three times four plus eight. So we've gone ahead and we've written out this entire sum right here. And you can find each of these pieces individually. Now you can crunch each of these numbers within the brackets yourself if you'd like to, and I'll just show you what these come out to in this video. So this is all gonna come out to six. And if you crunch these numbers, that'll come out to six as well. You need to square this, multiply three and a two, add eight to that result. Then you can run these numbers and that's going to give you eight, and you can run those numbers which will give you 12. So we have six plus six plus eight plus 12, which all comes out to 32 as our final result. So 32 is gonna be the solution to the sum, and that is how we can solve this problem. Hope you found this video helpful. Let's move on and get some more practice.

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:
$\sum_{i=1}^4i^2-3i+8$

Verified step by step guidance

Step 1: Understand the summation notation. The given summation is $ \sum_{i=1}^4 (i^2 - 3i + 8) $, which means you need to evaluate the expression $ i^2 - 3i + 8 $ for each integer value of $ i $ from 1 to 4, and then sum the results.

Step 2: Break the summation into individual terms. Substitute $ i = 1, 2, 3, 4 $ into the expression $ i^2 - 3i + 8 $ to calculate each term. For example, when $ i = 1 $, the term becomes $ 1^2 - 3(1) + 8 $. Repeat this for $ i = 2, 3, 4 $.

Step 3: Simplify each term. For each value of $ i $, simplify the expression $ i^2 - 3i + 8 $ to get a numerical result. For example, when $ i = 1 $, simplify $ 1^2 - 3(1) + 8 $ to get the first term. Do the same for $ i = 2, 3, 4 $.

Step 4: Add the results of the individual terms. Once you have the simplified values for each term (from $ i = 1 $ to $ i = 4 $), add them together to compute the total summation.

Step 5: Verify your work. Double-check each substitution, simplification, and addition to ensure accuracy in your calculations.