document.write( "Question 1197013: If f(x) = 9^x/(3 + 9^x), prove that:
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Algebra.Com's Answer #830080 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "If f(x) = 9^x/(3 + 9^x), prove that:
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document.write( "As first step, let's prove that f(x) + f(1-x) = 1  for any value of x.\r\n" );
document.write( "We have \r\n" );
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document.write( "    f(1-x) = by the definition of function f(x) =  = \r\n" );
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document.write( "           =  =  =  =  = .\r\n" );
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document.write( "    THEREFORE,  f(x) + f(1-x) =  +  =  = 1,\r\n" );
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document.write( "    and the statement is proved.\r\n" );
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document.write( "As the next step, let's write two identical sums in direct and inverse order\r\n" );
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document.write( "    f(1/2016)     + f(2/2016)    + f(3/2016)    + . . . + f(2015/2016)\r\n" );
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document.write( "    f(20125/2016) + f(2014/2016) + f(2013/2016) + . . . + f(1/2016)\r\n" );
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document.write( "and add them. Pairing the addends vertically, we have 2015 pairs of the form   + , \r\n" );
document.write( "and the sum in each such a pair equals 1.\r\n" );
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document.write( "So, the doubled sum equals 2015 and the sum itself is  ,  exactly as the problem states.\r\n" );
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