How to Find the Sum of Integers

Khan Academy Presents: Learn how to find the sum of integers.

Welcome to the presentation on finding sums of integers. You probably wondering why are we doing this within the context of averages and well if you think about it all in average is, is you take a sum of a bunch of numbers and you divide by the numbers the number of numbers you have and what we are going to do here is do a couple of algebra problems that involve just the sum parts first and actually they can carry over into, I guess, average problems as well. Let’s get started with the problem. Let’s say that I told you that I had the sum, let’s see the sum of five consecutive integers, consecutive integers is equal to 200. What is the smallest—I apologize for my hand writing. What is the smallest of the five integers? Well there’s a couple of ways to do this and but I guess the, the most straight forward way is to do it algebraically, I would say. So let’s say that X is the smallest of the integers right so X is actually what we’re going to want to figure out, smallest of integers. Well if X is the smallest what are the other four going to be right, we have a total of five, well they’re consecutive, consecutive just means that they follow each other like 5, 6, 7, 8, 9, 10. All of those are consecutive integers right and if you remember integers are just whole numbers, so it can’t be a fraction or a decimal. So if X is the smallest so then the next integer is going to be X plus one and then the one after that it’s going to be X plus two and the one after that is going to be X plus three and the one after that is going to be X plus four right. And it might seem confusing I’m writing all of these X’s but if you think about it if X was five, then this would be six this would be seven, this would be eight and this would be nine and that’s all I’m writing here right. So these would be assuming that X is the smallest of the integers, the five integers would be X, X plus 1, X plus 2, X plus 3, and X plus 4. And we know that the sum of these five consecutive integers is 200 well—what is the sum of these five I guess we could say numbers of expressions? Well let’s see we have 5 X’s 1 2 3 4 5 so X plus X plus X pus X plus X is equal to just 5X right? Or we could just say 5 times X and that’s plus one plus two is three, three plus three is six, six plus four is 10. So the sum of these five integers is 5X plus 10 and all I did is add up the X’s and then added up the constant and we know that is going to equal 200. And now this is just a level 2 linear equation we could just solve for X. So we get 5X is equal to 190 I just subtracted 10 from both sides right and then X is equal to let me divide five into 190, 5 goes into, let’s see five goes into 19 three times, three times five is 15, 9 minus 5 is 4 bring down the zero five goes into 48 times. So X is equal to 38, pretty straight forward problem don’t you think. Now what if I were to ask you what is the average of the five consecutive numbers, well now there’s two ways of doing this now that we already know that X is 38 we know that the other numbers are going to be, well this is 38, 39, 40, 41, 42. Well we could just average these four numbers you could just say 38 plus 39 plus 40 plus 41 plus 42 and well we already know—I think you don’t have to do the math, we already know that they average up –they sum up to 200 and then we divide the sum by five because there are five numbers so the average is 40. No not 46 the average is 40. And there’s a couple of ways you could think about that one you see 40 is just a middle number so that make sense and, and the only time we can really say is the middle numbers when the numbers are distributed evenly around 40 right? If we have a number that was much smaller than 40 or something, you couldn’t necessarily pick on the number but in this case this is consecutive and it make sense. Another way we could have done this problem if you were say taking the SCT and they were to ask you well you know the sum of five numbers is 200, what’s the averages of the number? Well you say well all I have to do is divide the 200 by five and I’ll get 40. Let’s do another problem, now I’ll make it a little bit harder, let’s say the sum of let’s say seven odd numbers, the sum of seven odd numbers let me make up a good I hope this one works I’m going to try to do it in my head, Is 200 and 17. What is the largest number alright I shouldn’t say number, seven odd integers. Actually it becomes a much harder problem if it was seven odd. But actually the only things that could be odder integers anyway so you can almost assume it. But the sum of seven odd integers is 217 what is the largest of the integers as you can tell them doing this on the fly. And actually my wife just diagnosed me, she thinks I have benign vertigo I got very dizzy this morning when I went to work so you have to forgive me for that as well that’s, that’s impairing me even more. Well let’s do this problem, let’s say that X is the largest. Then what would the number right below X be, would it be X minus 1? Well if X is an odd number, X minus 1 would be an even number so if we say that, if we say that—so in order to get the number right below which gives you X minus two to get another odd number. On my apologies, it should say the sum of seven consecutive odd. I don’t know if you assumed that. I’m trying my best today to confuse you. So the sum of seven consecutive odd integers is 217. What is the largest of the integers? So if X is the largest then the next smallest one would be X minus 2 right? Because it’s consecutive odd numbers not just consecutive right so consecutive odd numbers are like 1 3 5 7. You’re skipping the evens right so that’s why you’re going up or down by 2 depending how you view it. So the next one down will be X minus two, then we’ll have X minus four, X minus six, X minus eight X minus 10, X minus 12. I think that’s it 1 2 3 4 5 6 7 right those are seven numbers, they’re separated by two. X is the largest of them right, and we can assume that they’re odd because apparently the problem work out so that they’re odd. So what is the sum of these seven numbers well the 7 X’s just add up to 7X and then let’s see, let’s see two and four is six, six and six is 12, 12 and 8 is 20, 20 and 10 is 30, 30 and 12 is 32 so 7X minus 32 is equal to 217. And we just solve for X. 7X is equal to—see if we add 32 to both sides of these equation we get 249. Let’s see seven goes into 249. Is that right? It’s 200 right. So 7 goes into 249. Did I do this addition properly? I want to make sure, two plus four is six, six plus six is 12, 12 plus eight is 20, 20 plus 10 is 30, 30 plus 12 is 42. Oh here we go, See my mathematical spider sense could tell that something was fishy about this so that’s 7X minus 42 and that’s, so if we add 42 to both sides, it’s 7X is equal to 259 see how brave I am I do this thing in real time 259 so seven goes into 259. Let’s see seven goes into 25 three times, three times seven is 21 49 it goes into 37 times. So we get X is equal to 37 and we’re done. So just a review because I think I kind, I kind of got a lot of errors in this problem when I presented it. The question was the sum of seven consecutive odd integers is 217. What is the largest of the integers? I said the X is the largest and then if X is the largest the next smaller one will be X minus 2 because I’m not saying just consecutive integers, we’re saying consecutive odd integers right? So if X is 37 which is what we solved for then X minus two is 35. This is 33, this is 31, this is 29, this is 27, this is 25 and then we just add it up all the X’s and then solve, and then I’ll add it up all the constants and said well they add up to 217 and then we just solved for X. I think you’re now ready to try some of these problems, have fun.