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Compute the sum and the limit of the sum as $n \rightarrow \infty.$$$\sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{2 i}{n}\right)^{2}+4\left(\frac{i}{n}\right)\right]$$

$\frac{10}{3}$

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Integration

Section 2

Sums and Sigma Notation

Integrals

Integration Techniques

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So let's every multiplying in this one over and And this getting rid of this power here so we can rewrite this as a summation of Isaac to one of two square That's four I squared over. We have any times and squared, so that's cute. And then we have plus four over end squared. Okay, now let's break this up into Susan Nations. So know this year that for an end are constants. So we can just put out outside okay, and then looking at the other 2.1. And this is Equation three. And this is equation too. So be reading them in terms of our equations we have for over and cute. And then we have, um, in the times and plus one times two n plus one over six. And then we have four over and sward and times and plus one well lets you. Okay, so for over six, that's 2/3 to have two or in councils, and we're left with two and squared plus three n plus one over £6 squared that you have three. It's weird now and then we have plus this end cancels and we have to and plus two over. And now let's multiply this by end squared. Brush it out. It's word, um, three and over three end to come by this into a single fraction. Okay, so multiplying out to we get four and squared plus 12 and plus one, but 600 plus six. Actually, this is an squared and over three and squared. And in combining the same terms, we have 10 and squared, plus six plus 12. That's 18 and plus 1/3 and sweat. Okay, so we get the full information or some analysis, find limits as an approaches infinity, some of us that we have an inspired. That's a harsh terms. So what's the white but at so we see that these two terms cancel and we're left with, um our limit is equal to 10/3

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