Find the sum of the even integers between 21 and 45.

Question

Accepted Answer

Hey everyone welcome back in this problem. We are asked to consider the integers between 32 and 58. And were asked what is the sum of the odd integers between these numbers? Alright so we're looking for a sum. So let's write out our general some we want to start at I equals one and we're gonna go all the way up to Big Ben of our term. And now the term A. N. We can write as an arithmetic sequence. Okay so we have A. N. And in general we can write this as a one. The first term A plus and - times D. The difference. Alright well In our situation can we feel what can we fill it? But we know we're starting at 32. Okay. And we want to consider only odd imagers. So the first imager were going to consider is going to be 33. Okay so that's after 32 but it's odd. So we know that a one. It's going to be 33. So we're gonna have 33 plus and -1. And then we have D. What is d The difference between our terms? But we want to consider just odd integers. Okay so we're considering every other integer and so the difference between them is going to be two. Alright so now we have our general arithmetic sequence here so we can write each term. The only thing we have left to do is figure out what big N is, how high does this I value go in order to consider all of the intruders were interested in. Well let's look at a. N equals 33 Plus and -1 times two. Okay so this is our final term. What is our final term equal to? Well we know we want to stop at 58. We're looking only at odd integers. So 58 isn't odd. So the odd integer before 58 is gonna be 57. So our final term A N. Is going to be 57 And this is going to be equal to Plus and -1 times two. Okay. We want to solve for N. Let me move the 33 over. We get 24 is equal to N - times two, Dividing by 2. 12 is equal to N -1 and adding one. N. is equal to 13. Okay. Alright. So now we have are some going from I all the way up to big end of 13. So let's go ahead and find what that sum is. The sum s is going to be equal to. Let's recall the formula for the some, it's going to be N over two times a one plus A. M. Okay. And that's in the case where we have an arithmetic sequence like this. Alright well we know N we just found it it's 13 over to. Okay. A one is our first term which we talked about is A. M. Is our final term which we also talked about being 57. So now multiplying adding up, we are going to get that. The sum is equal to 585. So the answer To the sum of all the odd integers between 32 and 58. Well, it's going to be C- 585. That's it for this one. I hope this video helped see you in the next one.

Find the sum of the even integers between 21 and 45.

Key Concepts

Identifying Even Integers

Arithmetic Sequence

Sum of an Arithmetic Sequence

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