document.write( "Question 854334: For any integer n > 1, nth term = 1000^n / n! . Then the sequence {nth term}
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Algebra.Com's Answer #515275 by Edwin McCravy(20060)\"\" \"About 
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For any integer n > 1, nth term = 1000^n / n! . Then the sequence {nth term}
\n" ); document.write( "(A) does not have a maximum
\n" ); document.write( "(B) attains maximum at exactly one value of n
\n" ); document.write( "(C) attains maximum at exactly two values of n
\n" ); document.write( "(d) attains maximum for infinitely many values of n
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document.write( "The first term is 1000, the second term is 500000, so the sequence starts\r\n" );
document.write( "out increasing.  Let's investigate to see for how many terms the sequence\r\n" );
document.write( "is strictly increasing.\r\n" );
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document.write( "It will increase as long as a term is greater that the preceding term.\r\n" );
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document.write( "So we set the nth term greater than the n-1st term:\r\n" );
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document.write( "\"1000%5En%2Fn%21%3E1000%5E%28n-1%29%2F%28n-1%29%21\"\r\n" );
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document.write( "Multiply both sides by the LCD = n!\r\n" );
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document.write( "\"1000%5En%3E1000%5E%28n-1%29%2An\"\r\n" );
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document.write( "Divide both sides by \"1000%5E%28n-1%29\"\r\n" );
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document.write( "\"1000%3En\"\r\n" );
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document.write( "So the 999th term is the last term for which the sequence increases.\r\n" );
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document.write( "The 999th term is \r\n" );
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document.write( "\"%281000%5E1000%29%2F%281000%21%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "Then the 1000th term is\r\n" );
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document.write( "\"%281000%5E1000%29%2F%281000%21%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "which is equal to the 999th term since the last factor is \"1000%2F1000\" is 1.\r\n" );
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document.write( "After the 1000th term we will be multiplying by a fraction less than 1, \r\n" );
document.write( "beginning with multiplying by \"1000%2F1001\" and so the sequence will\r\n" );
document.write( "begin to decrease with the 1001st term.\r\n" );
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document.write( "So the 999th and the 1000th terms, which are equal, are the largest\r\n" );
document.write( "terms of the sequence.  Therefore\r\n" );
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document.write( "the sequence attains maximum at exactly two values of n, 999 and 1000.\r\n" );
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document.write( "Answer: (C)\r\n" );
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document.write( "Edwin
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