document.write( "Question 959812: What is the rule of the table?\r
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\n" ); document.write( "y | 1 | 3 | 7 | 15 | 29 | 51 |\r
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Algebra.Com's Answer #586735 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The first differences are 3-1, 7-3, 15-7, 29-15, 51-29\r\n" );
document.write( "or 2, 4, 8, 14, 22\r\n" );
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document.write( "The second differences are 4-2, 8-4, 14-8, 22-14\r\n" );
document.write( "or 2, 4, 6, 8\r\n" );
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document.write( "The third differences are 4-2, 6-4, 8-6\r\n" );
document.write( "or 2, 2, 2\r\n" );
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document.write( "Since the third differences are all the same, the equation is a\r\n" );
document.write( "third degree polynomial equation.  Suppose the third dregree \r\n" );
document.write( "polynomial equation is: \r\n" );
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document.write( "y = Ax³+Bx²+Cx+D\r\n" );
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document.write( "Substituting the first four points:\r\n" );
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document.write( "Substituting x=1, y=1, \r\n" );
document.write( "1 = A(1)³+B(1)²+C(1)+D or  A+B+C+D = 1\r\n" );
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document.write( "Substituting x=2, y=3, \r\n" );
document.write( "3 = A(2)³+B(2)²+C(2)+D or  8A+4B+2C+D = 3\r\n" );
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document.write( "Substituting x=3, y=7, \r\n" );
document.write( "7 = A(3)³+B(3)²+C(3)+D or  27A+9B+3C+D = 7\r\n" );
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document.write( "Substituting x=4, y=15, \r\n" );
document.write( "15 = A(4)³+B(4)²+C(4)+D or  64A+16B+4C+D = 15\r\n" );
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document.write( "Solve the system of 4 equations in 4 unknowns:\r\n" );
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document.write( "and get (A,B,C,D) = (1/3,-1,8/3,-1)\r\n" );
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document.write( "So the equation is\r\n" );
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document.write( "\"y=expr%281%2F3%29n%5E3-n%5E2%2Bexpr%288%2F3%29n-1\"\r\n" );
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document.write( "[If you are taking calculus of finite differences using\r\n" );
document.write( "Stirling numbers, then this can be done that way, but I\r\n" );
document.write( "assume you are not in that course.] \r\n" );
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document.write( "Edwin
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