document.write( "Question 297622: 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 #214278 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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document.write( "Let the answer be x.\r\n" );
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document.write( "1, -2, 3, 4, -5, 6, 7, -8, 9, 10…\r\n" );
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document.write( "The signs go in groups of three +, -,  +,  so the last group of three\r\n" );
document.write( "less than 100 that have signs like that will end in a multiple of 3,\r\n" );
document.write( "which is the group +97, -98, +99, since 99 is the largest multiple of 99.\r\n" );
document.write( "The last term would be +100 because 100 would be the beginning of the next \r\n" );
document.write( "group of three that go +, -, +, so we have:\r\n" );
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document.write( "1 - 2 + 3 + 4 - 5 + 6 + 7 - 8 + 9 + 10 + … + 97 - 98 + 99 +100 = 1750.\r\n" );
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document.write( "Next we look at the second sequence:\r\n" );
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document.write( "1, 2, -3, 4, 5, -6, 7, 8, -9, 10…\r\n" );
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document.write( "The signs also go in groups of three +, +,  -,  so the last group of three\r\n" );
document.write( "less than 100 that have signs like that are +97, +98, -99, so the last\r\n" );
document.write( "term would be +100 because it would be the beginning of the next group\r\n" );
document.write( "of three that go +, +, -, so we have:\r\n" );
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document.write( "1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 + 10 + … + 97 + 98 - 99 + 100 = x.\r\n" );
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document.write( "Now let's add the two sequences together term by term:\r\n" );
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document.write( "1 - 2 + 3 + 4 - 5 + 6 +  7 - 8 + 9 + 10 + … +  97 - 98 + 99 + 100 =   1750  \r\n" );
document.write( "1 + 2 - 3 + 4 + 5 - 6 +  7 + 8 - 9 + 10 + … +  97 + 98 - 99 + 100 =      x\r\n" );
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document.write( "2 + 0 + 0 + 8 + 0 + 0 + 14 + 0 + 0 + 20 + … + 194 +  0 +  0 + 200 = 1750+x\r\n" );
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document.write( "The left side is the arithmetic sequence with \"d=6\" and \"a%5B1%5D=2\" \r\n" );
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document.write( "2 + 8 + 14 + 20 + ... + 194 + 200\r\n" );
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document.write( "To find the number of terms, n, we can do it by observing the\r\n" );
document.write( "groups of 3, but if you can't tell there would be \"99%2F3%2B1\" \r\n" );
document.write( "or 34 terms, then we can calculate the n=34 this way:\r\n" );
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document.write( "\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29d\"\r\n" );
document.write( "\"200=2%2B%28n-1%296\"\r\n" );
document.write( "\"198=6%28n-1%29\"\r\n" );
document.write( "\"33=n-1\"\r\n" );
document.write( "\"34=n\"\r\n" );
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document.write( "So\r\n" );
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document.write( "\"S%5Bn%5D=%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"\r\n" );
document.write( "\"S%5B34%5D=%2834%2F2%29%282%2B200%29\"\r\n" );
document.write( "\"S%5B100%5D=%2817%29%28202%29=3434\"\r\n" );
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document.write( "Therefore the equation\r\n" );
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document.write( "2 + 0 + 0 + 8 + 0 + 0 + 14 + 0 + 0 + 20 + … + 194 +  0 +  0 + 200 = 1750+x\r\n" );
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document.write( "becomes\r\n" );
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document.write( "\"3434=1750%2Bx\"\r\n" );
document.write( "\"1684=x\"\r\n" );
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document.write( "and the required sum is 1684.\r\n" );
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document.write( "Edwin
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