Find the sum of the first 60 positive even integers.

Question

Accepted Answer

Hey everyone in this problem we are asked to consider the integers between 63 and 95. Okay. We're asked to find what is the sum of the even integers between those numbers. Okay. Alright. Let's start by writing out our summation. Okay. We know that we're going to write i is equal to something. Let's start at one. And this is gonna go up to some value end of A. M. Okay. So this is just the general some that we have. How can we write a N in a kind of situation like this? Okay. We're in an arithmetic type of situation. Well, we can write A. N. is equal to a one Plus N - Times D. Okay, We'll call this formula now. What a one tells us is the first term. Okay, So we're starting with the first term and then we're gonna add the difference t every single time we move forward with another term. Okay, that's what that formula is saying to us. Now, let's kind of figure it out for this situation. All right. Now in our situation, What is a one going to be? But we're starting at 63 but we want to consider only even integers. Okay, so the first even integer after 63 is going to be 64. So our a one is going to be 64. We're gonna have plus and minus one. And what about d what is the difference d The difference is going to be between each of the editors were interested in? We're looking at even integers. Okay. So we're looking for every other number. So the difference between each of them is going to be two. All right. So we have our general general formula for A. M. What we need to determine now is what is N? How many times are we going to do this? How high is there? So I'm going to go Well, we know that we want to stop at 95. Okay, well 95 is odd. So we're gonna take the number before 95, That's gonna be the last even interested. So we're gonna have 94 as our final a. n. This is going to be equal to 64 plus n minus one times to our general formula for A. M. Moving the 64 over we get 30 is equal to N -1 times two. Now we're solving for N. Okay, we want to find n. What is the highest value that we go to when we take the index? Yeah, 15 is equal to N -1 and so N is equal to 16. Okay, so now instead of writing and up here we know this is going to be 16. So now we have all the information of our summation. We know where we start. We know where we end. We have the A N. The general formula for each term. So let's go ahead and find the sum. Now recall that the sum were an arithmetic type summation is given by and over to times a one plus A. M. Well in this case RN is 16. We just found that so we get 16/2. A one is our first term which we know is 64 And an is our last term which we know is 94. Okay. The first even and the last even digit between 63 and 95. And this is going to give us eight times 158 for a final sum of 1,264. Okay. Alright. So the sum of all of the even integers between 63 and 95 is going to be a 1,264. Thanks everyone for watching. See you in the next video.

Find the sum of the first 60 positive even integers.

Key Concepts

Even Integers

Arithmetic Sequence

Sum of an Arithmetic Series

Watch next