document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209928>Question 1209928</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f satisfies\r<BR>\n" );
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document.write( "f(a + b) = f(a) + f(b) - ab<BR>\n" );
document.write( "for all nonnegative integers a and b, and f(1) = 7.  Compute f(123).<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851071 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851071\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the given equation be<BR>\n" );
document.write( "$$f(a+b) = f(a) + f(b) - ab$$<BR>\n" );
document.write( "for all nonnegative integers $a$ and $b$, and $f(1) = 7$.\r<BR>\n" );
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document.write( "First, we find $f(2)$:<BR>\n" );
document.write( "$$f(2) = f(1+1) = f(1) + f(1) - (1)(1) = 7 + 7 - 1 = 13$$<BR>\n" );
document.write( "Next, we find $f(3)$:<BR>\n" );
document.write( "$$f(3) = f(2+1) = f(2) + f(1) - (2)(1) = 13 + 7 - 2 = 18$$<BR>\n" );
document.write( "Next, we find $f(4)$:<BR>\n" );
document.write( "$$f(4) = f(3+1) = f(3) + f(1) - (3)(1) = 18 + 7 - 3 = 22$$\r<BR>\n" );
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document.write( "Let's try to find a pattern.<BR>\n" );
document.write( "$f(1) = 7 = 1^2 + 6(1)$<BR>\n" );
document.write( "$f(2) = 13 = 2^2 + 6(2) + 1$<BR>\n" );
document.write( "$f(3) = 18 = 3^2 + 6(3) - 9 = 9+18 = 27$<BR>\n" );
document.write( "$f(3) = 18 = 3^2 + 6(3) = 9 + 9 = 18$<BR>\n" );
document.write( "$f(4) = 22 = 4^2 + 6(4) - 18 = 16+24-18=22$\r<BR>\n" );
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document.write( "We can guess that $f(x) = x^2 + 6x$.<BR>\n" );
document.write( "Let's check if this satisfies the given equation.<BR>\n" );
document.write( "$$f(a+b) = (a+b)^2 + 6(a+b) = a^2 + 2ab + b^2 + 6a + 6b$$<BR>\n" );
document.write( "$$f(a) + f(b) - ab = (a^2 + 6a) + (b^2 + 6b) - ab = a^2 + 6a + b^2 + 6b - ab$$<BR>\n" );
document.write( "These are not equal. Let's try to find a pattern.\r<BR>\n" );
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document.write( "Let $f(x) = Ax^2 + Bx$.<BR>\n" );
document.write( "$f(1) = A+B = 7$<BR>\n" );
document.write( "$f(2) = 4A + 2B = 13$<BR>\n" );
document.write( "Multiply the first equation by 2:<BR>\n" );
document.write( "$2A + 2B = 14$<BR>\n" );
document.write( "Subtract this from the second equation:<BR>\n" );
document.write( "$2A = -1$<BR>\n" );
document.write( "$A = -1/2$<BR>\n" );
document.write( "$B = 7 - A = 7 + 1/2 = 15/2$<BR>\n" );
document.write( "Thus $f(x) = -\frac{1}{2} x^2 + \frac{15}{2} x$.\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "$f(a+b) = -\frac{1}{2} (a+b)^2 + \frac{15}{2} (a+b) = -\frac{1}{2} (a^2 + 2ab + b^2) + \frac{15}{2} a + \frac{15}{2} b$<BR>\n" );
document.write( "$f(a) + f(b) - ab = (-\frac{1}{2} a^2 + \frac{15}{2} a) + (-\frac{1}{2} b^2 + \frac{15}{2} b) - ab = -\frac{1}{2} a^2 - \frac{1}{2} b^2 + \frac{15}{2} a + \frac{15}{2} b - ab$<BR>\n" );
document.write( "$f(a+b) = f(a) + f(b) - ab$<BR>\n" );
document.write( "So $f(x) = -\frac{1}{2} x^2 + \frac{15}{2} x$.\r<BR>\n" );
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document.write( "Now, compute $f(123)$:<BR>\n" );
document.write( "$f(123) = -\frac{1}{2} (123)^2 + \frac{15}{2} (123) = \frac{1}{2} (15(123) - 123^2) = \frac{123}{2} (15 - 123) = \frac{123}{2} (-108) = 123(-54) = -6642$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{-6642}$<BR>\n" );
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