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How many three terms APs are possible whose elements belongs to a
\n" ); document.write( "given set of natural numbers {1,2,3,4,5,.....,40} and common
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document.write( "Case 1: For the middle term less than or equal to 20:\r\n" );
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document.write( "If 2 is the middle term, there is 1 possibility for the 1st term, \r\n" );
document.write( "{1}\r\n" );
document.write( "If 3 is the middle term, there are 2 possibilities for the 1st term, \r\n" );
document.write( "{1,2}\r\n" );
document.write( "If 4 is the middle term, there are 3 possibilities for the 1st term, \r\n" );
document.write( "{1,2,3} \r\n" );
document.write( "...\r\n" );
document.write( "This pattern continues to:\r\n" );
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document.write( "If 20 is the middle term, there are 19 possibilities for the 1st term, \r\n" );
document.write( "{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}\r\n" );
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document.write( "However this pattern stops at 20, because if the middle term is 21\r\n" );
document.write( "the 1st term cannot be 1 because the common difference will be 20,\r\n" );
document.write( "which would make the 3rd term be 21+20=41 which cannot be the 3rd term.\r\n" );
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document.write( "So for case 1, we need the sum of the first 19 positive integers.\r\n" );
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document.write( "The formula for the sum of the first n positive integers is n(n+1)/2\r\n" );
document.write( "and so for case 1, this will be (19)(20)/2 = 380/2 = 190\r\n" );
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document.write( "So for middle terms greater than 21 we have another pattern.\r\n" );
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document.write( "Case 2:\r\n" );
document.write( "For any 2nd term n, 39 ≥ n ≥ 21, \r\n" );
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document.write( "If 39 is the middle term, there is 1 possibility for the 3rd term, \r\n" );
document.write( "{40}\r\n" );
document.write( "If 38 is the middle term, there are 2 possibilities for the 3rd term, \r\n" );
document.write( "{39,40}\r\n" );
document.write( "If 37 is the middle term, there are 3 possibilities for the 3rd term, \r\n" );
document.write( "{1,2,3} \r\n" );
document.write( "...\r\n" );
document.write( "This pattern continues to\r\n" );
document.write( "If 21 is the middle term, there are 19 possibilities for the 3rd term, \r\n" );
document.write( "{22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39}\r\n" );
document.write( "This pattern, of course, stops at 21, and the rest are case 1.\r\n" );
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document.write( "This is also the sum of the first 19 positive integers, which we have\r\n" );
document.write( "already calculated as 190.\r\n" );
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document.write( "So the answer is twice that or 380.\r\n" );
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document.write( "Edwin \n" );
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