document.write( "Question 206236: The sum of the first 100 terms of the sequence 1, -2, 3, 4, -5, 6, 7, -8, 9, 10… is 1750. The sum of the first 100 terms of the sequence 1, 2, -3, 4, 5, -6, 7, 8, -9, 10… is equal to\r
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Algebra.Com's Answer #155852 by Edwin McCravy(20056) $\"\"$ $\"About$

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The sum of the first 100 terms of the sequence 1, -2, 3, 4, -5, 6, 7, -8, 9, 10… is 1750. The sum of the first 100 terms of the sequence 1, 2, -3, 4, 5, -6, 7, 8, -9, 10… is equal to
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\n" ); document.write( "(A) 1684 (B) 1717 (C) 1783 (D) 1816 (E) None of these
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 \r\n" );
document.write( "Let the sum be N\r\n" );
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document.write( "Write the last few terms of the sequence with the given sum as well:\r\n" );
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document.write( "We observe from the first few terms that only \r\n" );
document.write( "the multiples of 3 are negative and all other \r\n" );
document.write( "terms positive, so 96 and 99 are negative and \r\n" );
document.write( "97, 98,and 100 on the right end are positive.\r\n" );
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document.write( "1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 + 10 + ... +95-96+97+98-99+100 =  N\r\n" );
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document.write( "Now we do the same with the sequence with the \r\n" );
document.write( "given sum. We observe from the first few that \r\n" );
document.write( "only numbers just before multiples of 3 are \r\n" );
document.write( "negative and all the others positive, so 95 \r\n" );
document.write( "and 98 are negative and the others on the right \r\n" );
document.write( "end are positive.\r\n" );
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document.write( "  1 -  2 +  3 +  4 -  5 +  6 +  7 -  8  ... -95+96+97-98+99+100 = 1750\r\n" );
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document.write( "Now we write the terms of that sum in reverse order.  It will then, of\r\n" );
document.write( "course, give the same sum of 1750.\r\n" );
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document.write( "100 + 99 - 98 + 97 + 96 - 95 + 96 - 95  ... + 6- 5+ 4+ 3- 2+  1 = 1750 \r\n" );
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document.write( "Now we add that term by term to the original equation:\r\n" );
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document.write( "  1+  2-  3+  4+  5-  6+  7+  8+ ... +95- 96+ 97+ 98- 99+100 =      N\r\n" );
document.write( "100+ 99- 98+ 97+ 96- 95+ 94- 93+ ... + 6-  5+  4+  3-  2+  1 =   1750\r\n" );
document.write( "---------------------------------------------------------------------\r\n" );
document.write( "101+101-101+101+101-101+101-101+ ...+101-101+101+101-101+101 = N+1750   \r\n" );
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document.write( "There are 100 terms, so the sum on the left consists of 33 groups of \r\n" );
document.write( "three terms each (101+101-101) plus one 101 term on the right end, so \r\n" );
document.write( "the left side amounts to 33x(101+101-101)+101 or 33x101 + 101 = 34x101\r\n" );
document.write( "= 3434\r\n" );
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document.write( "So the bottom equation becomes ---------->              3434 = N+1750\r\n" );
document.write( "Solve \r\n" );
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document.write( "3434 = N+1750\r\n" );
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document.write( "for N and we get N = 1684, which is choice (A).\r\n" );
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document.write( "Edwin

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