Evaluate the following summation: ∑ i = 0 2 2 i 3 \sum_{i=0}^2\frac{2i}{3}

This problem, we're asked to evaluate the following summation, and we have this sum right down here. Now to do this, what we're going to do is treat this as our function, and then we have this variable I, which is gonna start at zero and increment all the way up to our high index here, which is two. So let's go ahead and figure this out. Now first up, what I'm gonna do is take zero and plug it into our function here. So we're gonna have two times zero over three. Then that's going to be plus, and then we're going to plug in one to our function. So we're going to have two times one over three, and we're going to plug our two into the function. So we're going to have two times two over three. So this is going to be the result when we plug everything into our sum. Now from here, I can evaluate each one of these pieces. Now since I see that there's a zero being multiplied up here, this whole thing is gonna come out to zero. They're going to have plus two times one over three, which is gonna be the same thing as two over three. They're going to have plus, and we have two times two over three, which is gonna be four over three. So this is what we get. Now from here, what I'm going to do is I'm gonna add these results. So we'll have two plus four, which is gonna give us six over three, and six over three is the same thing as two. So adding these fractions, then reducing this gives us two, giving us the solution to our problem. So this right here would be the answer, and that is how you can evaluate this sum. Hope you found this video helpful.

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8. Definite Integrals

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Calculus

8. Definite Integrals

Riemann Sums

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Multiple Choice

Evaluate the following summation:
$\sum_{i = 0}^{2} \frac{2 i}{3}$

$6$

$\frac{2}{3}$

$2$

$\frac{4}{3}$

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Verified step by step guidance

Identify the summation expression: $ \sum_{i=0}^2 \frac{2i}{3} $. This means you will evaluate the expression $ \frac{2i}{3} $ for each integer value of $ i $ from 0 to 2 and then sum the results.

Calculate the expression for $ i = 0 $: Substitute $ i = 0 $ into $ \frac{2i}{3} $ to get $ \frac{2 \times 0}{3} = 0 $.

Calculate the expression for $ i = 1 $: Substitute $ i = 1 $ into $ \frac{2i}{3} $ to get $ \frac{2 \times 1}{3} = \frac{2}{3} $.

Calculate the expression for $ i = 2 $: Substitute $ i = 2 $ into $ \frac{2i}{3} $ to get $ \frac{2 \times 2}{3} = \frac{4}{3} $.

Sum the results from each calculation: Add $ 0 + \frac{2}{3} + \frac{4}{3} $ to find the total sum of the series.