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# 2003 AIME II problem 8

## Video transcript

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 seeing 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 would be a plus 2m and if we just kept going we what we care is all the way to the eighth term because we're gonna take the eighth 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 the third term we added two M's so the eighth term we're going to add seven M's one less than the term the first term we added zero amps so the eighth term here is going to be is going to be a plus a plus seven M 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 seven and 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 coded for here a times B is going to be equal to fourteen hundred and forty one thousand four hundred one thousand four hundred and forty we also know that a les 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 17 hunt 1716 716 we also know we also know that a plus 2 m a plus 2 m times V plus 2 n times B plus 2 n 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 7 M times B plus 7 n 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 ends and ends 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 under this in a new color it's going to be let's just let's just multiply these two binomials out a times B it's going to be a B that's a times B plus a times 7n so that's plus 7a n plus 7 M times B so plus 7 BM plus 7m x 7n 7m x 7n so that's plus 49 plus 49 MN 49 49 MN 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 fourteen hundred and forty so we're already making some headway this right here is going to be equal to fourteen hundred and forty 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 get a times B which is a B plus a times n which is a n 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 a B is 1440 we know that this is fourteen hundred and forty so let's see let's subtract 1440 from both sides of this equation I'll switch colors again so let's subtract fourteen hundred and forty I guess I didn't switch colors from both sides of this equation these guys obviously cancel out the left-hand side becomes a n plus B M plus M n is equal to let's see the thousands cancel out we have a 671 minus 40 let's see we have a 6-0 is six then we can borrow this is a six this is 11 11 minus 4 is 7 6 minus 4 is 2 1 minus 1 is 0 so a n 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 a B plus a times 2n so it's 2 a n plus 2 MB times the 2m times B so it's plus 2 BM plus 2 m times 2n so plus 4 plus 4 m and is equal to 1800 and 48 well same idea this AV this is 1440 we can subtract 1400 and 40 from both sides of this equation I'm running out of space here so I'm 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 to a n I'm going to write it over here so it doesn't look confusing we have 2a and plus 2 BM plus 4m n is equal to these obviously canceled out 848 - 440 is 408 is equal to 408 did I do that right 848 yeah f4 408 right and now both sides equations are divisible by 2 so let's divide both sides by 2 so we have if we divide everything by 2 you get a n plus BM plus 2m n is equal to is equal to 204 now these look pretty close it's actually if we view if we view this is kind of one variable and this is another seems like we can solve for MN and then we can solve for a n 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 a n plus BM is we just remote apply that by 7 we're gonna get this term right over here so let's do that let's do that let's solve for M n 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 going to subtract this from this equation so multiply it by negative 1 we have negative a n minus BM minus m n 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 2 MN minus MN is just going to be MN and then 204 - 276 that's negative negative 72 so we were able to solve for MN and that's useful for us because this expression here is 49 MN so this number right here this number right here is negative that number right there is negative 72 now what's a n plus BM well we know that this right here is negative 72 this right here is negative 72 so let's 72 to both sides of this equation so plus 72 we get on the left-hand side a and plus BM these cancel out R is equal to so 6 plus 2 is 8 7 plus 7 is 14 you have a 1 up there 1 plus 2 is 3 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 AV which is the same thing as fourteen hundred and forty 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 a n plus BM now we know that a n 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 do so 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 well we could just multiply it out so seven times 348 I have no calculator at my disposal seven times 348 8 times 7 is 56 4 times 7 is 28 plus 3 is or plus 5 is 33 3 times 7 is 21 plus 3 is 24 so this is 2436 if we add that to 1440 so if you add it to flip did I do that right 7 times 8 is 56 7 times 4 is 28 plus 5 is 20 33 yep that's right so that we add the 1440 over here we get six seven eight three three thousand 876 so that's this whole part over here that's all of this business and then we have to take 49 times 70 - and then subtract it from this so let's take 49 times 72 2 times 9 is 18 2 times 4 is 8 plus 1 is 9 stick a 0 here 9 times 7 is 63 7 times 4 is 28 plus 6 is 34 34 so this is 8 12 carry the 1 5 & 3 so we're gonna subtract 3528 so we get what do we get here so this becomes a 16 this becomes a 6 16 minus 8 is 8 6 minus 2 is 4 8 minus 5 is 3 and then 3 minus 3 is 0 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