document.write( "Question 512989: Evaluate :
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Algebra.Com's Answer #342845 by Edwin McCravy(20056)\"\" \"About 
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document.write( "1 + (1+2) + (1+2+3) + (1+2+3+4) + ··· + (1+2+···+100)\r\n" );
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document.write( "I'll get one more term:\r\n" );
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document.write( "1 + (1+2) + (1+2+3) + (1+2+3+4) + (1+2+3+4+5) + ··· + (1+2+···+100)\r\n" );
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document.write( "1 + 3 + 6 + 10 + 15 + ··· + (1+2+···+100)\r\n" );
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document.write( "n   an  Sn  d1   d2  d3\r\n" );
document.write( "1   1   1   1   2   1\r\n" );
document.write( "2   3   4   3   3   \r\n" );
document.write( "3   6  10   6      \r\n" );
document.write( "4  10  20     \r\n" );
document.write( "5  15     \r\n" );
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document.write( "It takes three differences to get a constant column, so\r\n" );
document.write( "we will assume a quadratic sum formula Sn\r\n" );
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document.write( "Sn = An³ + Bn² + Cn + D\r\n" );
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document.write( "S1 = A(1)³ + B(1)² + Cn + D\r\n" );
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document.write( "1 = A + B + C + D\r\n" );
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document.write( "S2 = A(2)³ + B(2)² + C(2) + D\r\n" );
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document.write( "4 = A(8) + B(4) + 2C + D\r\n" );
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document.write( "4 = 8A + 4B + 2C + D\r\n" );
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document.write( "S3 = A(3)³ + B(3)² + C(3) + D\r\n" );
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document.write( "10 = A(27) + 9B + 3C + D\r\n" );
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document.write( "10 = 27A + 9B + 3C + D\r\n" );
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document.write( "S4 = A(4)³ + B(4)² + C(4) + D\r\n" );
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document.write( "20 = A(64) + 16B + 4C + D\r\n" );
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document.write( "20 = 64A + 16B + 4C + D\r\n" );
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document.write( "So we have the system of equations:\r\n" );
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document.write( " 1 =   A +   B +  C + D\r\n" );
document.write( " 4 =  8A +  4B + 2C + D\r\n" );
document.write( "10 = 27A +  9B + 3C + D\r\n" );
document.write( "20 = 64A + 16B + 4C + D\r\n" );
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document.write( "or in standard order:\r\n" );
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document.write( "  A +   B +  C + D =  1\r\n" );
document.write( " 8A +  4B + 2C + D =  4\r\n" );
document.write( "27A +  9B + 3C + D = 10\r\n" );
document.write( "64A + 16B + 4C + D = 20\r\n" );
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document.write( "Solve that system and get\r\n" );
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document.write( "A = \"1%2F6\", B = \"1%2F2\", C = \"1%2F3\", D = 0\r\n" );
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document.write( "Then this\r\n" );
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document.write( "Sn = An³ + Bn² + Cn + D\r\n" );
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document.write( "becomes\r\n" );
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document.write( "Sn = \"1%2F6\"n³ + \"1%2F2\"n² + \"1%2F3\"n + D\r\n" );
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document.write( "Sn = \"1%2F6\"n³ + \"3%2F6\"n² + \"2%2F6\"n + D\r\n" );
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document.write( "Sn = \"expr%281%2F6%29\"(n³+3n²+2n)\r\n" );
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document.write( "Sn = \"expr%281%2F6%29\"n(n²+3n+2)\r\n" );
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document.write( "Sn = \"expr%28n%2F6%29\"(n+1)(n+2)\r\n" );
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document.write( "That's the general formula, so the sum of the first 100 terms is\r\n" );
document.write( "found by substituting n = 100:\r\n" );
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document.write( "S100 = \"expr%28100%2F6%29\"(100+1)(100+2)\r\n" );
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document.write( "S100 = \"expr%2850%2F3%29\"(101)(102)\r\n" );
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document.write( "S100 = 171700\r\n" );
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document.write( "Edwin
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