document.write( "Question 823053: While adding the first few continuous natural numbers, a candidate missed one of the numbers and wrote the answer as 177. What was the number missed? \n" ); document.write( "
Algebra.Com's Answer #496084 by Edwin McCravy(20056)\"\" \"About 
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While adding the first few continuous natural numbers, a candidate missed one of the numbers and wrote the answer as 177. What was the number missed?
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document.write( "The formula for the sum of the first n natural numbers is \"n%28n%2B1%29%2F2\"\r\n" );
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document.write( "Let the number missed be = k  \r\n" );
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document.write( "\"n%28n%2B1%29%2F2\"-k = 177\r\n" );
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document.write( "\"%28n%5E2%2Bn%29%2F2\"-k = 177\r\n" );
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document.write( "Multiply both sides by 2\r\n" );
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document.write( "nē + n - 2k = 354 \r\n" );
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document.write( "nē + n - 354 = 2k\r\n" );
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document.write( "k ≧ 1, so 2k ≧ 2\r\n" );
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document.write( "nē + n - 354 ≧ 2\r\n" );
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document.write( "nē + n - 352 ≧ 0\r\n" );
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document.write( "By the quadratic formula, that has critical\r\n" );
document.write( "numbers approximately 18.3 and -19.3\r\n" );
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document.write( "In integers the inequality has solutions \r\n" );
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document.write( "n = -20,-21,-23,...  and n = 19, 20, 21,...\r\n" );
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document.write( "Since n is positive we take the first integer \r\n" );
document.write( "value exceeding 18.3, which is 19:\r\n" );
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document.write( "Substituting n=19 in\r\n" );
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document.write( "\"n%5E2%2F2\" + \"n%2F2\" - 177 = k\r\n" );
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document.write( "\"19%5E2%2F2\" + \"19%2F2\" - 177 = k\r\n" );
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document.write( "180.5 + 9.5 - 177 = k\r\n" );
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document.write( "13 = k\r\n" );
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document.write( "Answer:  the candidate added the integers 1 through 19, but omitted 13.\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "So since n = 19, the candidate added this with 13 omitted:\r\n" );
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document.write( "1+2+3+4+5+6+7+8+9+10+11+12+14+15+16+17+18+19 = 177 \r\n" );
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document.write( "That checks.\r\n" );
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document.write( "Edwin
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