document.write( "Question 368945: 9, 19, 40, ?, 146 \n" ); document.write( "
Algebra.Com's Answer #263037 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are two methods.  \r\n" );
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document.write( "First method.  Differences:\r\n" );
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document.write( "list them with an x where the ? is\r\n" );
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document.write( "  9 \r\n" );
document.write( " 19 \r\n" );
document.write( " 40 \r\n" );
document.write( "  x \r\n" );
document.write( "146\r\n" );
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document.write( "Beside each of the the first four numbers write the difference between\r\n" );
document.write( "the next number below it.  That is,\r\n" );
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document.write( "Beside the 9 write 19-9 = 10\r\n" );
document.write( "Beside the 19 write 40-19 = 21\r\n" );
document.write( "Beside the 40 write x-40\r\n" );
document.write( "Beside the x write 146-x which is 146-x\r\n" );
document.write( "The numbers in the second column below are called \"the first differences\".\r\n" );
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document.write( "  9      10\r\n" );
document.write( " 19      21\r\n" );
document.write( " 40    x-40\r\n" );
document.write( "  x   146-x \r\n" );
document.write( "146\r\n" );
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document.write( "Do the same as before:\r\n" );
document.write( "Beside the 10 write 21-10 = 11\r\n" );
document.write( "Beside the 21 write (x-40)-21 = x-40-21 = x-61\r\n" );
document.write( "Beside the x-40 write (146-x)-(x-40) = 146-x-x+40 = 186-2x\r\n" );
document.write( "The numbers in the third column below are called \"the second differences\".\r\n" );
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document.write( "  9      10       11\r\n" );
document.write( " 19      21     x-61\r\n" );
document.write( " 40    x-40   186-2x \r\n" );
document.write( "  x   146-x \r\n" );
document.write( "146\r\n" );
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document.write( "Beside the 11 write (x-61)-11 = x-61-11 = x-72\r\n" );
document.write( "Beside the x-61 write (186-2x)-(x-61) = 186-2x-x+61 = 247-3x\r\n" );
document.write( "The numbers in the fourth column below are called \"the third differences\".\r\n" );
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document.write( "  9      10       11      x-72\r\n" );
document.write( " 19      21     x-61    247-3x\r\n" );
document.write( " 40    x-40   186-2x \r\n" );
document.write( "  x   146-x \r\n" );
document.write( "146\r\n" );
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document.write( "Beside the x-72 write (247-3x)-(x-72) = 247-3x-x+72 = 319-4x\r\n" );
document.write( "There is only one fourth differences.  We set it equal to 0.\r\n" );
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document.write( "  9      10       11      x-72   319-4x\r\n" );
document.write( " 19      21     x-61    247-3x\r\n" );
document.write( " 40    x-40   186-2x \r\n" );
document.write( "  x   146-x \r\n" );
document.write( "146\r\n" );
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document.write( "Set 319-4x = 0\r\n" );
document.write( "       -4x = -319\r\n" );
document.write( "         x = 319/4 \r\n" );
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document.write( "That's the missing value.\r\n" );
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document.write( "-----------------------------\r\n" );
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document.write( "Second method, find the polynomial equation for \"a%5Bn%5D\"\r\n" );
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document.write( "  9 = a1  \r\n" );
document.write( " 19 = a2 \r\n" );
document.write( " 40 = a3 \r\n" );
document.write( "  x = a4 \r\n" );
document.write( "146 = a5\r\n" );
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document.write( "Assume \"An%5E3+%2BBn%5E2%2BCn%2BD=a%5Bn%5D\"\r\n" );
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document.write( "Substitute \"n=1\" and \"a%5B1%5D=9\"\r\n" );
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document.write( "\"A%281%29%5E3+%2BB%281%29%5E2%2BC%281%29%2BD=a%5B1%5D\"\r\n" );
document.write( "\"A+%2BB%2BC%2BD=9\"\r\n" );
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document.write( "Substitute \"n=2\" and \"a%5B2%5D=19\"\r\n" );
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document.write( "\"A%282%29%5E3+%2BB%282%29%5E2%2BC%282%29%2BD=a%5B2%5D\"\r\n" );
document.write( "\"8A+%2B4B%2B2C%2BD=19\"\r\n" );
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document.write( "Substitute \"n=3\" and \"a%5B3%5D=40\"\r\n" );
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document.write( "\"A%283%29%5E3+%2BB%283%29%5E2%2BC%283%29%2BD=40\"\r\n" );
document.write( "\"27A+%2B9B%2B3C%2BD=40\"\r\n" );
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document.write( "Substitute \"n=5\" and \"a%5B5%5D=146\"\r\n" );
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document.write( "\"A%285%29%5E3+%2BB%285%29%5E2%2BC%285%29%2BD=146\"\r\n" );
document.write( "\"125A+%2B25B%2B5C%2BD=146\"\r\n" );
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document.write( "So we have the system of equations:\r\n" );
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document.write( "Solve that and get \"A=31%2F24\", \"B=-9%2F4\", \"C=185%2F24\", \"D=9%2F4\"\r\n" );
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document.write( "So \r\n" );
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document.write( "\"An%5E3+%2BBn%5E2%2BCn%2BD=a%5Bn%5D\"\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "So the missing term a4 is found by substituting n=4\r\n" );
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document.write( "\"248%2F3-36%2B185%2F6%2B9%2F4=a%5B4%5D\"\r\n" );
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document.write( "\"992%2F12-432%2F12%2B370%2F12%2B27%2F12=a%5B4%5D\"\r\n" );
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document.write( "\"957%2F12=a%5B4%5D\"\r\n" );
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document.write( "\"319%2F4=a%5B4%5D\"\r\n" );
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document.write( "Edwin
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