document.write( "Question 1012259: So.. for my holiday homework i was told to find a formula to sum up the tetrahedral numbers.. which are 1, 4, 10, 20, 35, 56...\r
\n" ); document.write( "\n" ); document.write( "And im really stuck on where to start, could you please help me step by step to sum up consecutive tetrahedral numbers? (Would appreciate if its in a easy mathematical language)\r
\n" ); document.write( "\n" ); document.write( "Is there any tricks to find formulas or equstions from a sequence?\r
\n" ); document.write( "\n" ); document.write( "Thank you :\"c \n" ); document.write( "
Algebra.Com's Answer #628142 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "1, 4, 10, 20, 35, 56...\r\n" );
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document.write( "There is no one way that always works with every sequence.\r\n" );
document.write( "But I always try a difference table first.   \r\n" );
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document.write( "List the given numbers in a column:\r\n" );
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document.write( " 1\r\n" );
document.write( " 4\r\n" );
document.write( "10\r\n" );
document.write( "20\r\n" );
document.write( "35\r\n" );
document.write( "56\r\n" );
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document.write( "Subtract every number from the one directly below it, and write the \r\n" );
document.write( "result out beside each number creating a second column of numbers:\r\n" );
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document.write( " 1   3\r\n" );
document.write( " 4   6\r\n" );
document.write( "10  10 \r\n" );
document.write( "20  15 \r\n" );
document.write( "35  21\r\n" );
document.write( "56\r\n" );
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document.write( "Do that with the second column and continue this procedure until all\r\n" );
document.write( "the numbers in a column are the same:\r\n" );
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document.write( " 1   3   3   1\r\n" );
document.write( " 4   6   4   1\r\n" );
document.write( "10  10   5   1\r\n" );
document.write( "20  15   6 \r\n" );
document.write( "35  21\r\n" );
document.write( "56 \r\n" );
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document.write( "Since it takes 3 columns after the first to get them all the same, \r\n" );
document.write( "that means that a polynomial formula for the kth term must be of degree 3.\r\n" );
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document.write( "So we assume that the nth term is of the form:\r\n" );
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document.write( "\"a%5Bk%5D=A%2Ak%5E3%2BB%2Ak%5E2%2BC%2Ak%2BD\"\r\n" );
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document.write( "So we substitute k=1,2,3,4 and the given numbers in that and get this \r\n" );
document.write( "system of equations:\r\n" );
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document.write( "which reduces to \r\n" );
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document.write( "Eliminate the D's by subtracting the \r\n" );
document.write( "1st equation from the 2nd,\r\n" );
document.write( "2nd equation from the 3rd,\r\n" );
document.write( "3rd equation from the 4th,\r\n" );
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document.write( "\"system%28%0D%0A7A%2B3B%2BC=3%2C%0D%0A19A%2B5B%2BC=6%2C%0D%0A37A%2B7B%2BC=10%29\"\r\n" );
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document.write( "Eliminate the C's by subtracting the \r\n" );
document.write( "1st equation from the 2nd,\r\n" );
document.write( "2nd equation from the 3rd,\r\n" );
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document.write( "\"system%28%0D%0A12A%2B2B=3%2C%0D%0A18A%2B2B=4%29\"\r\n" );
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document.write( "Eliminate the B's by subtracting the \r\n" );
document.write( "1st equation from the 2nd:\r\n" );
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document.write( "\"6A=1\"\r\n" );
document.write( "\"A=1%2F6\"\r\n" );
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document.write( "Substitute in \r\n" );
document.write( "\"12A%2B2B=3\" \r\n" );
document.write( "\"12%281%2F6%29%2B2B=3\"\r\n" );
document.write( "\"2%2B2B=3\"\r\n" );
document.write( "\"2B=1\"\r\n" );
document.write( "\"B=1%2F2\"\r\n" );
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document.write( "Substitute in \r\n" );
document.write( "\"7A%2B3B%2BC=3\"\r\n" );
document.write( "\"7%281%2F6%29%2B3%281%2F2%29%2BC=3\"\r\n" );
document.write( "\"7%2F6%2B3%2F2%2BC=3\"\r\n" );
document.write( "Multiply through by LCD = 6 to clear the fractions\r\n" );
document.write( "\"7%2B9%2B6C=18\"\r\n" );
document.write( "\"16%2B6C=18\"\r\n" );
document.write( "\"6C=2\"\r\n" );
document.write( "\"C=1%2F3\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "\"A%2BB%2BC%2BD=1\"\r\n" );
document.write( "\"1%2F6%2B1%2F2%2B1%2F3%2BD=1\"\r\n" );
document.write( "Multiply through by LCD = 6 to clear the fractions\r\n" );
document.write( "\"1%2B3%2B2%2B6D=6\"\r\n" );
document.write( "\"6%2B6D=6\"\r\n" );
document.write( "\"6D=0\"\r\n" );
document.write( "\"D=0\"\r\n" );
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document.write( "So the formula is:\r\n" );
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document.write( "\"a%5Bk%5D=expr%281%2F6%29k%5E3%2Bexpr%281%2F2%29k%5E2%2Bexpr%281%2F3%29k\"\r\n" );
document.write( "Factor out \"1%2F6\"\r\n" );
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document.write( "\"a%5Bk%5D=expr%281%2F6%29%28k%5E3%2B3k%5E2%2B2k%29\"\r\n" );
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document.write( "So \r\n" );
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document.write( "\"sum%28%28a%5Bk%5D%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\"sum%28%28+expr%281%2F6%29%28k%5E3%2B3k%5E2%2B2k%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\"expr%281%2F6%29sum%28+%28k%5E3%2B3k%5E2%2B2k%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "\"%22%22=%22%22\"\"%22%22=%22%22\"\r\n" );
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document.write( "We have formulas for all three of those sums:\r\n" );
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document.write( "\"sum%28%28k%5E3%29%2Ck=1%2Cn%29=expr%281%2F4%29n%5E2%28n%2B1%29%5E2\"\r\n" );
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document.write( "\"sum%28%28k%5E2%29%2Ck=1%2Cn%29=expr%281%2F6%29n%28n%2B1%29%282n%2B1%29\"\r\n" );
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document.write( "\"sum%28%28k%29%2Ck=1%2Cn%29=expr%281%2F2%29n%28n%2B1%29\"\r\n" );
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document.write( "So we now have:\r\n" );
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document.write( "\"%22%22=%22%22\"\r\n" );
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document.write( "\"%22%22=%22%22\"\r\n" );
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document.write( "Get an LCD inside the big parentheses:\r\n" );
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document.write( "\"%22%22=%22%22\"\r\n" );
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document.write( "Factor \"1%2F4\" out of the big parentheses:\r\n" );
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document.write( "\"%22%22=%22%22\"\r\n" );
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document.write( "Simplify the fraction in front and factor n(n+1) out of the big\r\n" );
document.write( "parentheses:\r\n" );
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document.write( "\"%22%22=%22%22\"\r\n" );
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document.write( "Remove the little parentheses inside the big parentheses:\r\n" );
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document.write( "\"expr%281%2F24%29n%28n%2B1%29%28n%5E2%2Bn%2B4n%2B2%5E%22%22%2B4%5E%22%22%29%5E%22%22\"\"%22%22=%22%22\"\r\n" );
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document.write( "Combine terms inside:\r\n" );
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document.write( "\"expr%281%2F24%29n%28n%2B1%29%28n%5E2%2B5n%2B6%5E%22%22%29%5E%22%22\"\"%22%22=%22%22\"\r\n" );
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document.write( "Factor the quadratic trinomial:\r\n" );
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document.write( "\"expr%281%2F24%29n%28n%2B1%29%28n%2B2%29%28n%2B3%29\"\r\n" );
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document.write( "Edwin
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