document.write( "Question 1116736: What is the nth term rule of the quadratic sequence below?
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Algebra.Com's Answer #731650 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The good lady mistakenly thought it was an arithmetic \r\n" );
document.write( "sequence, but it is a quadratic sequence, so we assume:\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5Bn%5D%2C%22%22=%22%22%2CAn%5E2%2BBn%2BC%29\"\r\n" );
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document.write( "[Don't confuse the capital A with the subscripted small an,\r\n" );
document.write( "for the nth term.  They are different.]\r\n" );
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document.write( "There are 3 unknowns A,B, and C, so we need 3 equations,\r\n" );
document.write( "so we only need the first 3 terms, with n=1,2,3.\r\n" );
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document.write( "\"a%5B1%5D=6\", \"a%5B2%5D=12\", \"a%5B3%5D=20\"\r\n" );
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document.write( "Substitute n=1:\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5B1%5D%2C%22%22=%22%22%2CA%281%29%5E2%2BB%281%29%2BC%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0A6%2C%22%22=%22%22%2CA%2BB%2BC%29\"\r\n" );
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document.write( "So the first equation is \"A%2BB%2BC=6\"\r\n" );
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document.write( "Substitute n=2:\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5B2%5D%2C%22%22=%22%22%2CA%282%29%5E2%2BB%282%29%2BC%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0A12%2C%22%22=%22%22%2C4A%2B2B%2BC%29\"\r\n" );
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document.write( "So the second equation is \"4A%2B2B%2BC=12\"\r\n" );
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document.write( "Substitute n=3:\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5B3%5D%2C%22%22=%22%22%2CA%283%29%5E2%2BB%283%29%2BC%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0A20%2C%22%22=%22%22%2C9A%2B3B%2BC%29\"\r\n" );
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document.write( "So the third equation is \"9A%2B3B%2BC=20\"\r\n" );
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document.write( "We have this 3×3 system:\r\n" );
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document.write( "\"system%28A%2BB%2BC=6%2C4A%2B2B%2BC=12%2C9A%2B3B%2BC=20%29\"\r\n" );
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document.write( "Subtract the first equation from the second:\r\n" );
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document.write( "\"3A%2BB=6\"\r\n" );
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document.write( "Subtract the second equation from the third equation:\r\n" );
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document.write( "\"5A%2BB=8\"\r\n" );
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document.write( "So we have this 2×2 system:\r\n" );
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document.write( "\"system%283A%2BB=6%2C5A%2BB=8%29\"\r\n" );
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document.write( "Subtract the first equation from the second:\r\n" );
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document.write( "\"2A=2\"\r\n" );
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document.write( "\"A=1\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "\"3A%2BB=6\"\r\n" );
document.write( "\"3%281%29%2BB=6\"\r\n" );
document.write( "\"3%2BB=6\"\r\n" );
document.write( "\"B=3\"\r\n" );
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document.write( "Substitute in \r\n" );
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document.write( "\"A%2BB%2BC=6\"\r\n" );
document.write( "\"1%2B3%2BC=6\"\r\n" );
document.write( "\"4%2BC=6\"\r\n" );
document.write( "\"C=2\"\r\n" );
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document.write( "We substitute in the expression for the nth term \r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5Bn%5D%2C%22%22=%22%22%2CAn%5E2%2BBn%2BC%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5Bn%5D%2C%22%22=%22%22%2C1n%5E2%2B3n%2B2%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5Bn%5D%2C%22%22=%22%22%2Cn%5E2%2B3n%2B2%29\"    \r\n" );
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document.write( "If you like you can factor that as\r\n" );
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document.write( "\"matrix%281%2C3%2C%0D%0A%0D%0Aa%5Bn%5D%2C%22%22=%22%22%2C%28n%2B1%29%28n%2B2%29%29\"    \r\n" );
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document.write( "Checking to make sure it works for n=1,...,7 with the given sequence:\r\n" );
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document.write( "Edwin
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