Algebra.Com's Answer #833161 by ikleyn(52788)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "The increasing sequence T = 2 3 5 6 7 8 10 11 consists of all positive integers
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document.write( " that are not perfect squares. What is the 2012th term of T?
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document.write( " This problem has an underwater stone like a trap.\r
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document.write( " We should be careful in order for not to fall into the trap.\r
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document.write( " In the course of my solution, I will show you where the trap is \r
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document.write( " and what to do to avoid falling into the trap.\r
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document.write( " = 44.86 (rounded).\r\n" );
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document.write( "So, the original sequence was the sequence of all natural numbers, starting from 1\r\n" );
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document.write( " 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, . . . \r\n" );
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document.write( "from which 44 terms (perfect squares) were excluded.\r\n" );
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document.write( "Hence, one can think that the 2012-th term of the given sequence is \r\n" );
document.write( "the (2012 + 44)-th term of the sequence of all natural numbers, starting from 1,\r\n" );
document.write( "which is the number (2012 + 44) =2056.\r\n" );
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document.write( " | Here is the trap. To avoid falling into the trap, we should check |\r\n" );
document.write( " | if there is a perfect unaccounted square between  and 2056. | \r\n" );
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document.write( "So, we check the next perfect square  = 2025, and we see that \r\n" );
document.write( "such a perfect square does really exist.\r\n" );
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document.write( "Therefore, we make a correction to the previous estimate and conclude that \r\n" );
document.write( "the 2012-th term is the number 2057, one unit greater than the previous estimate of 2056.\r\n" );
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document.write( "ANSWER. The 2012-th term of the described sequence is 2057.\r\n" );
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document.write( "Solved.\r
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