document.write( "Question 290530: Given that 1, 3, 7, … is a linear recurrence sequence, find its closed form.
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Algebra.Com's Answer #210250 by Edwin McCravy(20067)\"\" \"About 
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Given that 1, 3, 7, … is a linear recurrence sequence, find its closed form.
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document.write( "List the three given terms:                          1   3   7\r\n" );
document.write( "List the successive differences                        2   4\r\n" );
document.write( "List the successive diferences of diferences             2\r\n" );
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document.write( "It takes 2 differences to get to a row\r\n" );
document.write( "containing only one number, so we assume\r\n" );
document.write( "a polynomial in n of degree 2. \r\n" );
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document.write( "\"a%5Bn%5D=An%5E2%2BBn%2BC\"\r\n" );
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document.write( "For n = 1\r\n" );
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document.write( "\"a%5B1%5D=A%281%29%5E2%2BB%281%29%2BC\"\r\n" );
document.write( "\"1=A%281%29%5E2%2BB%281%29%2BC\"\r\n" );
document.write( "\"1=A%2BB%2BC\"\r\n" );
document.write( "\"A%2BB%2BC=1\"\r\n" );
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document.write( "For n = 1\r\n" );
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document.write( "\"a%5B2%5D=A%282%29%5E2%2BB%282%29%2BC\"\r\n" );
document.write( "\"3=A%282%29%5E2%2B2B%2BC\"\r\n" );
document.write( "\"3=A%284%29%2B2B%2BC\"\r\n" );
document.write( "\"3=4A%2B2B%2BC\"\r\n" );
document.write( "\"4A%2B2B%2BC=3\"\r\n" );
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document.write( "For n = 3\r\n" );
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document.write( "\"a%5B3%5D=A%283%29%5E2%2BB%283%29%2BC\"\r\n" );
document.write( "\"7=A%283%29%5E2%2B3B%2BC\"\r\n" );
document.write( "\"7=A%289%29%2B3B%2BC\"\r\n" );
document.write( "\"7=9A%2B3B%2BC\"\r\n" );
document.write( "\"9A%2B3B%2BC=7\"\r\n" );
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document.write( "So we have the system of equations:\r\n" );
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document.write( "\"system%28A%2BB%2BC=1%2C+4A%2B2B%2BC=3%2C9A%2B3B%2BC=7%29\"\r\n" );
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document.write( "Solve that and get A=1, B=-1, C=1\r\n" );
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document.write( "Substituting in:\r\n" );
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document.write( "\"a%5Bn%5D=An%5E2%2BBn%2BC\"\r\n" );
document.write( "\"a%5Bn%5D=%281%29n%5E2%2B%28-1%29n%2B%281%29\"\r\n" );
document.write( "\"a%5Bn%5D=n%5E2-n%2B1\"\r\n" );
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document.write( "Edwin
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