Beg Algebra: Solving for Consecutive Numbers

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SOLVING FOR CONSECUTIVE NUMBERS
Okay, well now we’re going to look at one of my all time favorite word problems, and it’s so much fun and I really like it so much that I don’t even want to contaminate it with some sort stupid little humor thing, because this is just too good. It’s like insulting the flag. This is great.
Let’s take a look at this, because this is a classic and some of my favorites. So you want to find three consecutive odd natural numbers, whose sum is 99. This is great. Now, first of all, what do consecutive odd numbers mean? Well, it means that odd numbers that are as close together as possible. So, for example, when you think of the numbers lining up here and you forget about the evens, you have this 5. And then right after 5 there’s 7. That’s the next odd number, so those are consecutive, because we’re just forgetting 6, 6 is even. And then we forget 9, but then, of course, 9 would be the next one. So these are three consecutive odd numbers. And if you add them up, well, we don’t get 99. And the question is what are the three consecutive odd numbers that, if you add them up, you get 99? That’s the question at hand.
Well, so the first question is how do you actually represent these odd numbers? One thing that you can do and a lot of people, in fact, think about this, is they say, “Well, gee, I’ll call the first number x, the next one y, the next one z.” So then x + y + z = 99. Well, that’s fine; the only problem there is that you have all those unknowns. How are you going to figure out what all those unknowns are? We have to somehow use the fact that, first of all, these numbers are odd and, second of all, they’re consecutive, because those are actually important facts that have to come into play here.
So what does it mean to be odd? If someone tells you you're odd, what do you know? Well, it means that you're just one more than an even number. And the even numbers are always 2 times something. So, in fact, if you’re an odd number, then you know that you must have the following shape: you must have the basic shape 2 times something plus 1. That’s how every single odd number looks. For example, notice that 5 is actually 2(2) + 1. And 7 is actually what? Well, that would be, well, it’s the number, so I would just take 2(2) + 3, because it’s the next odd number so I have to add 2 more, 5, 6, 7. So I add two more to this. And the 9 can be written as 2(2) + 5, because to get from 5 to 9 I have to add 5 – 1, 2, 3, 4, 5. So, if I’m thinking of three consecutive odd numbers, they will always have this shape: 2 times something plus 1, and then the next on will be 2 times something plus 3, and the next one will be 2 times something plus 5. That’s always the shape of three consecutive odd numbers. I don’t know what this thing is in general, so we’ll just call that s for now. But once you know s, then I could find the first number, the second number and the third number, just like I did here with the 5, with the 7 and with the 9.
Okay, well now forget about 5, 7 and 9. Let’s think about this in general, because we know this is not the answer. 5 + 7 + 9 ? 99. So let’s just let the first odd number be 2s + 1. If the first odd number is 2s + 1, then what’s the next consecutive odd number? Well, it would be 2s + 3. So that’s the next odd number, it’s two away from the previous one. And then what’s the next odd number after that? I would be 2s + 5. So I’d have 2s + 5. Great! Those are three consecutive odd numbers. They’re all odd and they’re all consecutive, there’s no odd numbers between them.
And what do I want to do? I want to add them all up and what’s the answer? The answer is 99. But I can actually add these up. Let’s add them up! Here I have 5 and 3 is 8, and 1 is 9. And here I have 2s + 2s + 2s, so that’s 6s. So I have 6s + 9 and that has to equal what? Well, what’s the sum? If you look at the question, the sum is 99.
All right, well now I want to solve here for s. So what does s equal? Well, 6s equals – if I bring this 9 over and I subtract, I get 90. And so therefore what does s equal? Well, s = 15, if I divide both sides by 6.
Okay s = 15, so that means the number are 15, 17 and 19, right? No, no, and no. This is s. If you want to find out the numbers, we have to go back to here. This is a great, great mistake to make. You get so excited that you see an odd number that you just jump at it. Well, you can’t always jump at the odd number that you see up there. You have to think about what it means. Remember the odd numbers we’re looking at have the form 2s + 1, 2s + 3 and 2s + 5. Another way you could have detected this, by the way, is if you would have checked your answer. If you take 15, add it to 17 and add it to 19, you're far, far away from 99. So be careful there.
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Okay, but if we go back to this number, what I see is that this number is now going to be 15(2) + 1, which is 31, which forces this number to be 33, which forces this number to be 35. Notice those are three consecutive odd numbers and voile! If you add them all up, we get 99. Neato! I just love these questions!
Anyway, I hope you enjoy them and realize that the key to it is to realize how to write down three consecutive odd numbers in a row and add them up. Of course, you can imagine similar questions with even numbers. And what does it mean to be a consecutive even number? You would have 2s, then 2s + 2, and then 2s + 4; same exact idea.
Enjoy them and I hope you like them as much as I do. I just love them! Aren’t they fun? And, by the way, you could juggle, and in fact, I’ll juggle for you right now. What? We’re out of film. Sorry.