document.write( "Question 1196482: A function f is defined for integers m and n as given: f(mn)=f(m)f(n)-f(m+n)+ 1001, where either m or n is equal to 1, and f(1)=2.\r
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Algebra.Com's Answer #829325 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "A function f is defined for integers m and n as given: f(mn)=f(m)f(n)-f(m+n)+ 1001, where either m or n is equal to 1, and f(1)=2.
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\n" ); document.write( "b) Find the value of f(9999).
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document.write( "In the given formula, take x = m, n = 1.  You will get\r\n" );
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document.write( "    f(x*1)  = f(x)*f(1) - f(x+1) + 1001,  or\r\n" );
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document.write( "    f(x)    = 2*f(x)    - f(x+1) + 1001,  which is the same as\r\n" );
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document.write( "     0      =   f(x)    - f(x+1) + 1001,\r\n" );
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document.write( "     f(x+1) =   f(x) + 1001.                   (1)\r\n" );
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document.write( "Thus part (a) is just solved and completed.\r\n" );
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document.write( "From the formula (1), we conclude that the sequense (1) is an arithmetic progression \r\n" );
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document.write( "with the first term f(1) = 2 and the common difference of 1001.  So\r\n" );
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document.write( "    f(9999) = f(1) + (9999-1)*1001 = 2 + (9999-1)*1001 = 10,008,000.\r\n" );
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document.write( "Thus part (b) is completed, too.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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