document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209945>Question 1209945</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f has the following properties:<BR>\n" );
document.write( "* f(a,b) is defined for all positive integers a and b\r<BR>\n" );
document.write( "\n" );
document.write( "* f(a,1) = a\r<BR>\n" );
document.write( "\n" );
document.write( "* f(a,b) = 1 if b > a\r<BR>\n" );
document.write( "\n" );
document.write( "* f(a + 1,b) = b[f(a,b) - f(a,b - 1)]\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Compute f(4,1) + f(4,2) + f(4,3) + f(4,4).<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851156 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=851156\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's compute the values of f(4,1), f(4,2), f(4,3), and f(4,4) using the given properties.\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **f(a,1) = a**<BR>\n" );
document.write( "    * f(4,1) = 4\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **f(a + 1, b) = b[f(a,b) - f(a,b - 1)]**\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **f(a,b) = 1 if b > a**\r<BR>\n" );
document.write( "\n" );
document.write( "Now, let's calculate the required values:\r<BR>\n" );
document.write( "\n" );
document.write( "* **f(4,1) = 4** (given)\r<BR>\n" );
document.write( "\n" );
document.write( "* **f(4,2):**<BR>\n" );
document.write( "    * f(4,2) = 2[f(3,2) - f(3,1)]<BR>\n" );
document.write( "    * f(3,1) = 3<BR>\n" );
document.write( "    * f(3,2) = 2[f(2,2) - f(2,1)]<BR>\n" );
document.write( "    * f(2,1) = 2<BR>\n" );
document.write( "    * f(2,2) = 2[f(1,2) - f(1,1)]<BR>\n" );
document.write( "    * f(1,1) = 1<BR>\n" );
document.write( "    * f(1,2) = 1 (since 2 > 1)<BR>\n" );
document.write( "    * f(2,2) = 2[1 - 1] = 0<BR>\n" );
document.write( "    * f(3,2) = 2[0 - 2] = -4<BR>\n" );
document.write( "    * f(4,2) = 2[-4 - 3] = 2[-7] = -14\r<BR>\n" );
document.write( "\n" );
document.write( "* **f(4,3):**<BR>\n" );
document.write( "    * f(4,3) = 3[f(3,3) - f(3,2)]<BR>\n" );
document.write( "    * f(3,2) = -4<BR>\n" );
document.write( "    * f(3,3) = 3[f(2,3) - f(2,2)]<BR>\n" );
document.write( "    * f(2,2) = 0<BR>\n" );
document.write( "    * f(2,3) = 1 (since 3 > 2)<BR>\n" );
document.write( "    * f(3,3) = 3[1 - 0] = 3<BR>\n" );
document.write( "    * f(4,3) = 3[3 - (-4)] = 3[7] = 21\r<BR>\n" );
document.write( "\n" );
document.write( "* **f(4,4):**<BR>\n" );
document.write( "    * f(4,4) = 4[f(3,4) - f(3,3)]<BR>\n" );
document.write( "    * f(3,3) = 3<BR>\n" );
document.write( "    * f(3,4) = 1 (since 4 > 3)<BR>\n" );
document.write( "    * f(4,4) = 4[1 - 3] = 4[-2] = -8\r<BR>\n" );
document.write( "\n" );
document.write( "Now, sum the values:\r<BR>\n" );
document.write( "\n" );
document.write( "f(4,1) + f(4,2) + f(4,3) + f(4,4) = 4 + (-14) + 21 + (-8) = 4 - 14 + 21 - 8 = 3\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, f(4,1) + f(4,2) + f(4,3) + f(4,4) = 3.\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{3}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );