document.write( "Question 1131575: What is the value of the product (1-1/2^2)(1-1/3^2)(1-1/4^2)...(1-1/999^2)? \n" ); document.write( "
Algebra.Com's Answer #748251 by ikleyn(52933)\"\" \"About 
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document.write( "First, it is clear that\r\n" );
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document.write( "    P = \"%281-1%2F2%5E2%29%281-1%2F3%5E2%29%281-1%2F4%5E2%29%2Aellipsis%2A%281-1%2F999%5E2%29\" = \r\n" );
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document.write( "         \"%281-1%2F2%29%2A%281-1%2F3%29%2A%281-1%2F4%29%2Aellipsis%2A%281-1%2F999%29\"*(\"%281%2B1%2F2%29%2A%281%2B1%2F3%29%2A%281%2B1%2F4%29%2Aellipsis%2A%281%2B1%2F999%29\"\r\n" );
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document.write( "Now, consider  Q = \"%281-1%2F2%29%2A%281-1%2F3%29%2A%281-1%2F4%29%2Aellipsis%2A%281-1%2F999%29\".\r\n" );
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document.write( "    Notice that the denominator of the first fraction is the numerator of the second fraction.\r\n" );
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document.write( "    Next, the denominator of the second fraction is the numerator of the third fraction.\r\n" );
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document.write( "    This pattern continues: the denominator of the third fraction is the numerator of the fourth fraction.\r\n" );
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document.write( "    And so on . . . \r\n" );
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document.write( "    Cancel all common factors in numerators and denominators to get the value  Q = \"1%2F999\".\r\n" );
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document.write( "Next, consider  R = \"%281%2B1%2F2%29%2A%281%2B1%2F3%29%2A%281%2B1%2F4%29%2Aellipsis%2A%281%2B1%2F999%29\".\r\n" );
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document.write( "    Notice that the numerator of the first fraction is the denominator of the second fraction.\r\n" );
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document.write( "    Next, the numerator of the second fraction is the denominator of the third fraction.\r\n" );
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document.write( "    This pattern continues: the numerator of the third fraction is the denominator of the fourth fraction.\r\n" );
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document.write( "    And so on . . . \r\n" );
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document.write( "    Cancel all common factors in numerators and denominators to get the value  R = \"%281%2F2%29%2A1000\" = 500.\r\n" );
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document.write( "So, the final answer is  P = Q*R = \"%281%2F999%29%2A500\" = \"500%2F999\".    ANSWER.\r\n" );
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