document.write( "Question 898860: A series of 384 consecutive odd integers has a sum that is a perfect fourth power of a positive integer. Find the smallest possible sum for this series.\r
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Algebra.Com's Answer #545050 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let the first odd integer be 2k-1 where k is an integer\r\n" );
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document.write( "The sum of an arithmetic series is\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29\"\r\n" );
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document.write( "With odd integers d=2, we'll let a1 = 2k-1, n=384\r\n" );
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document.write( "\"S%5B384%5D=expr%28384%2F2%29%282%282k-1%29%2B%28384-1%292%5B%22%22%5D%29\"\r\n" );
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document.write( "\"S%5B384%5D=192%282%282k-1%29%2B%28383%292%5B%22%22%5D%29\"\r\n" );
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document.write( "\"S%5B384%5D=2%5E6%2A3%284k-2%2B766%29\"\r\n" );
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document.write( "\"S%5B384%5D=2%5E6%2A3%284k%2B764%29\"\r\n" );
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document.write( "\"S%5B384%5D=2%5E6%2A3%2A4%28k%2B191%29\"\r\n" );
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document.write( "\"S%5B384%5D=2%5E6%2A3%2A2%5E2%2A%28k%2B191%29\"\r\n" );
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document.write( "\"S%5B384%5D=2%5E8%2A3%28k%2B191%29\"\r\n" );
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document.write( "\"2%5E8\" is a perfect 4th power. Since we have one factor of 3,\r\n" );
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document.write( "the smallest 4th power possible will be \"2%5E8%2A3%5E4\".\r\n" );
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document.write( "That's the answer: \r\n" );
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document.write( "\"2%5E8%2A3%5E4=256%2A81=20736=12%5E4\", choice b\r\n" );
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document.write( "---------\r\n" );
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document.write( "Following through,\r\n" );
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document.write( "This will be when \"k%2B191+=3%5E3\" \r\n" );
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document.write( "\"k%2B191+=27%7D%7D%0D%0A%0D%0A%7B%7B%7Bk=-164\"\r\n" );
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document.write( "Notice that the problem just said \"consecutive odd integers\", it didn't\r\n" );
document.write( "say they had to be positive!\r\n" );
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document.write( "So the sequence of 384 odd integers starts with 2k-1 = 2(-164)-1 = -329\r\n" );
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document.write( "and ends with \"a%5B384%5D=-329%2B%28384-1%29%282%29=437\"\r\n" );
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document.write( "So it's the series of 384 consecutive odd integers\r\n" );
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document.write( "(-329)+(-327)+...+435+437 = 20736 = 124\r\n" );
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document.write( "Edwin
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