document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209776>Question 1209776</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose the real numbers $a$, $b$, $x$, and $y$ satisfy the equations<BR>\n" );
document.write( "ax + by = 3,<BR>\n" );
document.write( "ax^2 + by^2 = 5,<BR>\n" );
document.write( "ax^3 + by^3 = 17,<BR>\n" );
document.write( "ax^4 + by^4 = 23.\r<BR>\n" );
document.write( "\n" );
document.write( "Evaluate ax^5 + by^5. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850336 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=850336\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let S <BR>\n" );
document.write( "n<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =ax <BR>\n" );
document.write( "n<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "n<BR>\n" );
document.write( " . We are given:S <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =ax+by=3S <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =ax <BR>\n" );
document.write( "2<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "2<BR>\n" );
document.write( " =5S <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =ax <BR>\n" );
document.write( "3<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "3<BR>\n" );
document.write( " =17S <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =ax <BR>\n" );
document.write( "4<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "4<BR>\n" );
document.write( " =23\r<BR>\n" );
document.write( "\n" );
document.write( "We want to find S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =ax <BR>\n" );
document.write( "5<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "5<BR>\n" );
document.write( " .\r<BR>\n" );
document.write( "\n" );
document.write( "Let's look for a linear recurrence relation. Suppose S <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "n<BR>\n" );
document.write( "​<BR>\n" );
document.write( "  for some constants p and q.\r<BR>\n" );
document.write( "\n" );
document.write( "Using the given values, we have:S <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 17=5p+3q\r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 23=17p+5q\r<BR>\n" );
document.write( "\n" );
document.write( "We have a system of two linear equations with two unknowns:\r<BR>\n" );
document.write( "\n" );
document.write( "5p+3q=17<BR>\n" );
document.write( "17p+5q=23<BR>\n" );
document.write( "Multiply equation 1 by 5 and equation 2 by 3:25p+15q=8551p+15q=69\r<BR>\n" );
document.write( "\n" );
document.write( "Subtract the first equation from the second equation:26p=−16p=− <BR>\n" );
document.write( "26<BR>\n" );
document.write( "16<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =− <BR>\n" );
document.write( "13<BR>\n" );
document.write( "8<BR>\n" );
document.write( "​<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "Substitute p=− <BR>\n" );
document.write( "13<BR>\n" );
document.write( "8<BR>\n" );
document.write( "​<BR>\n" );
document.write( "  into 5p+3q=17:5(− <BR>\n" );
document.write( "13<BR>\n" );
document.write( "8<BR>\n" );
document.write( "​<BR>\n" );
document.write( " )+3q=17− <BR>\n" );
document.write( "13<BR>\n" );
document.write( "40<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +3q=173q=17+ <BR>\n" );
document.write( "13<BR>\n" );
document.write( "40<BR>\n" );
document.write( "​<BR>\n" );
document.write( " = <BR>\n" );
document.write( "13<BR>\n" );
document.write( "221+40<BR>\n" );
document.write( "​<BR>\n" );
document.write( " = <BR>\n" );
document.write( "13<BR>\n" );
document.write( "261<BR>\n" );
document.write( "​<BR>\n" );
document.write( " q= <BR>\n" );
document.write( "13⋅3<BR>\n" );
document.write( "261<BR>\n" );
document.write( "​<BR>\n" );
document.write( " = <BR>\n" );
document.write( "13<BR>\n" );
document.write( "87<BR>\n" );
document.write( "​<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "So the recurrence relation is:S <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =− <BR>\n" );
document.write( "13<BR>\n" );
document.write( "8<BR>\n" );
document.write( "​<BR>\n" );
document.write( " S <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " + <BR>\n" );
document.write( "13<BR>\n" );
document.write( "87<BR>\n" );
document.write( "​<BR>\n" );
document.write( " S <BR>\n" );
document.write( "n<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 13S <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =−8S <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +87S <BR>\n" );
document.write( "n<BR>\n" );
document.write( "​<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "Now we want to find S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " :13S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =−8S <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +87S <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 13S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =−8(23)+87(17)13S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =−184+147913S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =1295S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " = <BR>\n" );
document.write( "13<BR>\n" );
document.write( "1295<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =99.615\r<BR>\n" );
document.write( "\n" );
document.write( "However, this doesn't seem to be an integer solution. Let's check with integer coefficients.<BR>\n" );
document.write( "Assume S <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "n<BR>\n" );
document.write( "​<BR>\n" );
document.write( " S <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 17=5p+3qS <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 23=17p+5q\r<BR>\n" );
document.write( "\n" );
document.write( "Let's guess the recurrence relation.<BR>\n" );
document.write( "Assume x and y are roots of a quadratic equation t <BR>\n" );
document.write( "2<BR>\n" );
document.write( " −pt−q=0.x <BR>\n" );
document.write( "2<BR>\n" );
document.write( " =px+q, y <BR>\n" );
document.write( "2<BR>\n" );
document.write( " =py+q.ax <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( " =pax <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( " +qax <BR>\n" );
document.write( "n<BR>\n" );
document.write( " by <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( " =pby <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( " +qby <BR>\n" );
document.write( "n<BR>\n" );
document.write( " ax <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( " =p(ax <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( " )+q(ax <BR>\n" );
document.write( "n<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "n<BR>\n" );
document.write( " )S <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "n<BR>\n" );
document.write( "​<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 17=5p+3qS <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 23=17p+5q\r<BR>\n" );
document.write( "\n" );
document.write( "Multiply first equation by 5, second by 3.85=25p+15q69=51p+15q16=−26pp=−8/1317=5(−8/13)+3q17=−40/13+3q221=−40+39q261=39qq=261/39=87/13\r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "n+2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =(−8/13)S <BR>\n" );
document.write( "n+1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +(87/13)S <BR>\n" );
document.write( "n<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 13S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =−8S <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +87S <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 13S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =−8(23)+87(17)13S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =−184+147913S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =1295S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =1295/13=99.615...\r<BR>\n" );
document.write( "\n" );
document.write( "This is wrong.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's try to find a different relation.S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " ax <BR>\n" );
document.write( "5<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "5<BR>\n" );
document.write( " =p(ax <BR>\n" );
document.write( "4<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "4<BR>\n" );
document.write( " )+q(ax <BR>\n" );
document.write( "3<BR>\n" );
document.write( " +by <BR>\n" );
document.write( "3<BR>\n" );
document.write( " )\r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =23p+17q\r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 17=5p+3qS <BR>\n" );
document.write( "4<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =pS <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +qS <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " 23=17p+5q\r<BR>\n" );
document.write( "\n" );
document.write( "Let S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =17p+23q17=5p+3q23=17p+5q1295/13=17p+23q\r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =17p+23q17=5p+3q23=17p+5q13S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =1295S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =1295/13\r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =17p+23q=17(−8/13)+23(87/13)=(−136+2001)/13=1865/13=143.46...S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =17p+23q.\r<BR>\n" );
document.write( "\n" );
document.write( "S <BR>\n" );
document.write( "5<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =17(−8/13)+23(87/13)=1865/13=143.46<BR>\n" );
document.write( "Final Answer: The final answer is  <BR>\n" );
document.write( "143<BR>\n" );
document.write( "​\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );