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2003 AIME II problem 8

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- [Voiceover] Find the eighth term of the sequence 1440, 1716, 1848, and so on and so forth, whose terms are formed by multiplying the corresponding terms of two arithmetic sequences. Arithmetic sequences, always having trouble saying that properly. So, let's think about the two sequences and then we'll take the products of corresponding terms to get the terms of this sequence. So maybe one of the arithmetic sequences starts with A, starts with A, and then the next one, the next term in that sequence is going to be some constant plus A, that's what an arithmetic sequence is. It's just a constant added to each successive term. So, it's going to be A, A plus M, and then the next term would be M plus A plus M, or it'd be A plus two M. And if we just kept going, what we care is all the way to the eighth term because we're gonna take the eight term of this sequence, multiply it by the eighth term of the other sequence to get the eighth term of this sequence over here. So, the second term, we only added one M, the third term, we added two M's, so the eighth term, we're gonna add seven M's, one less than the term. The first term, we added zero M's. So, the eighth term here is going to be, is going to be A plus, A plus 7m. That's the eighth term of this arithmetic sequence. Now, let's do another one for another arithmetic sequence, maybe that starts at the number B. Now, the next term is going to be some constant plus B, so let's just call that B plus N. Then the third term will be N plus B plus N. So, it'll be B plus 2n. And then we go all the way to the eighth term by the same logic, it's going to be B plus, B plus 7n. The difference between each of these terms is N. Now, they tell us that the product of the corresponding terms of these two sequences are the terms of this sequence. so, A times B, so when you take the product of A and B, we get A times B, A times, I'll color code it for here, A times B is going to be equal to 1440. 1400, 1440. We also know that A plus M, we also know that A plus M times B plus N, times B plus N is going to be equal to, is going to be equal to 1716. 1716. We also know, we also know that A plus 2m, A plus 2m times B plus 2n, times B plus 2n is going to be equal to 1848. Is going to be equal to 1848. And if we want the eighth term of this sequence, we just have to take this product right over here. So, the eighth term is going to be, is going to be A plus 7m, A plus 7m times B plus 7n. Times B plus, B plus 7n. Now, before I even do this, it seems like we have some good information to at least put some constraints around our A's, B's, M's, and N's. Let's just think about the form that this is going to take, and then maybe we can kind of build up the numbers as we solve for this stuff over here. So, the eighth term in our sequence is going to be, I'll do this in a new color, it's going to be, let's just multiply these two binomials out, A times B, it's going to be AB, that's A times B, plus A times 7n, so that's plus 7an, plus 7m times B, so plus 7bm, plus 7m times 7n. 7m times 7n, so that's plus 49, plus 49mn, 49, 49mn. So, this is what we need to figure out the value of. We need to figure out the value of this right here. Well, we already know one of the terms here. We already know that A times b is equal to 1440. So, we're already making some head way. This right here is going to be equal to 1440. Now, let's see if we can figure out anything for these terms over here using the other information over here. So, let's multiply these out. So, let me do this once again in a new color. So, if I multiply this out, I got A times B, which is AB plus A times N, which is AN, plus M times B, or B times M, which is BM, plus M times N, plus M times N. Which is going to be equal to 1716. Now, we know that AB is 1440. We know that this is 1440. So let's see, let's subtract 1440 from both sides of this equation, I'll switch colors again. So, let's subtract 1440. Well, I guess I didn't switch colors. >From both sides of this equation, these guys obviously cancel out, the left hand side becomes AN plus BM plus MN is equal to, let's see, the thousands cancel out, we have a 671 minus 40, let's see, we have a six minus zero is six. Then we can borrow, this is a six, this is 11, 11 minus four is seven, six minus four is two, one minus one is zero. So, AN plus BM plus MN is equal to 276. So, we got that information using this and that, that, essentially, equation. Now let's use this equation over here. Do this in a new color, I'm running out of colors. I'll do it in green. So, we have A times B, which is AB plus A times 2n. So, it's 2an. Plus 2mb times, 2m times B, so it's plus 2bm. Plus 2m times 2n, so plus four, plus 4mn is equal to 1848. Well, same idea, this AB, this is 1440. We can subtract 1440 from both sides of this equation. I'm running out of space here so I'm just gonna subtract 1440 from the left and the right side. Now this equation becomes, on the left hand side, we're just left with this stuff over here. So, we have 2an. I'm gonna write it over here so it doesn't look confusing. We have 2an plus 2bm plus 4mn is equal to, these obviously cancelled out. 848 minus 440 is 408. Is equal to 408, did I do that right? 848, yeah, 408, right. And now both sides of this equation are divisible by two, so let's divide both sides by two. So, we have, if we divide everything by two, you get AN pus BM plus 2mn is equal to, is equal to 204. Now, these look pretty close. It's actually, if we view, if we view this as kind of one variable, and this as another, seems like we can solve for MN, and then we can solve for AN BM. And the reason why that looks useful, the reason why that would be useful is our final answer, we have some multiple times MN. So, if we know what MN is, we could put that number here, and if we know what AN plus BM is, we just multiply that seven, and we're gonna get this term right over here, so let's do that. Let's do that, let's solve for MN. And to solve for MN, let's subtract this equation from this equation. So, let me do it in that same color. So, I'm gonna subtract this from this equation. So, multiply it by negative one. We have negative AN minus BM minus MN is equal to negative 276, is equal to negative 276. And then when we subtract, we get, these cancel out, these cancel out, and then 2mn minus MN is just gonna be MN. And then 204 minus 276, that's negative, negative 72. So, we're able to solve for MN. And that's useful for us because this expression here is 49mn. So, this number right here, this number right here is negative, that number right there is negative 72. Now, what's AN plus BM? Well, we know that this right here is negative 72. This right here is negative 72. So, let's add 72 to both sides of this equation. So, plus 72. We get on the left hand side, AN plus BM, these cancel out, is equal to, so six plus two is eight, seven plus seven is 14, you have a one up there, one plus two is three. So, it's 348. So, the eighth term in our sequence, which we figured out, which we figured out was this business over here, it is equal to AB. Which is the same thing as 1440. Plus, plus this thing over here. Plus this thing, which is just the same thing as seven times, let me write it up here. This thing over here in blue is the same thing as seven times AN plus BM. Now, we know that AN plus BM is 348, so it's seven times 348. And then finally, finally, plus, plus this thing over here, plus 49 times MN. 49, or we could say minus 49 times 72. So, let me just, so what this is, you could say 49 times negative 72. This is going to be the eighth term in our sequence, so let's figure out, let's figure out what this thing, what this thing even is. So, let's see how we can, let's see how we can do it. Well, we could just multiply it out. So, seven times 348. I have no calculator at my disposal, seven times 348. Eight times seven is 56. Four times seven is 28 plus three is, or, plus five is 33. Three times seven is 21 plus three is 24. So this is 2436. If we add that to 1440, so, if you add it to, did I do that right? Seven times eight is 56, seven times four is 28, plus five is 33. Yup, that's right. So then we add the 1440 over here. We get six seven eight three, 3876. So, that's this whole part over here. That's all of this business. And then we have to take 49 times 72 and then subtract it from this. So, let's take 49 times 72. Two times nine is 18, two times four is eight, plus one is nine, stick a zero here. Nine times seven is 63. Seven times four is 28 plus six is 34. 34, so this is eight, 12, carry the one, five, and three. So, we're gonna subtract 3528. So, we get, what do we get here? So, this becomes a 16, this becomes a six, 16 minus eight is eight. Six minus two is four. Eight minus five is three and then three minus three is zero. So, the eighth term in this sequence, the eighth term in this sequence is going to be, I just wrote it down, it is 348. 348. And we're done.