document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195898>Question 1195898</A>: <I><SPAN STYLE=\"word-break: break-all;\"> https://artofproblemsolving.com/texer/japjtbbh\r<BR>\n" );
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document.write( "Suppose that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u_n ALIGN=MIDDLE  ALT=\"u_n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a sequence of real numbers satisfying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u_%28n%2B2%29=2u_%28n%2B1%29%2Bu_n ALIGN=MIDDLE  ALT=\"u_%28n%2B2%29=2u_%28n%2B1%29%2Bu_n\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u_3=9 ALIGN=MIDDLE  ALT=\"u_3=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u_6=128 ALIGN=MIDDLE  ALT=\"u_6=128\"   DISABLED_style= \"max-width:90%; height:auto;\" >. What is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u_5 ALIGN=MIDDLE  ALT=\"u_5\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\r<BR>\n" );
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document.write( "*Edited </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828513 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828513\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If the sequence is<BR>\n" );
document.write( "u_(n+1) = 2u_(n+1) + u_(n)<BR>\n" );
document.write( "then you will arrive at the conclusion that @ikleyn reached: in that the terms should alternate in sign. <BR>\n" );
document.write( "However, assuming that u3 = 9 and u6 = 128 are correct, then there's no way the above sequence works.\r<BR>\n" );
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document.write( "I noticed that your link mentioned this instead<BR>\n" );
document.write( "u_(n+2) = 2u_(n+1) + u_(n)\r<BR>\n" );
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document.write( "The n+1 portion at the front was changed to n+2\r<BR>\n" );
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document.write( "If that's the case, then we can say the following two equations<BR>\n" );
document.write( "u6 = 2*u5 + u4<BR>\n" );
document.write( "u5 = 2*u4 + u3\r<BR>\n" );
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document.write( "Let,<BR>\n" );
document.write( "x = u5<BR>\n" );
document.write( "y = u4<BR>\n" );
document.write( "and furthermore we are given,<BR>\n" );
document.write( "u3 = 9<BR>\n" );
document.write( "u6 = 128\r<BR>\n" );
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document.write( "So those two equations mentioned update to<BR>\n" );
document.write( "128 = 2*x + y<BR>\n" );
document.write( "x = 2*y + 9<BR>\n" );
document.write( "after plugging in the given knowns and unknowns.\r<BR>\n" );
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document.write( "Apply substitution to solve for y.<BR>\n" );
document.write( "128 = 2*x + y<BR>\n" );
document.write( "128 = 2*(2y+9) + y<BR>\n" );
document.write( "128 = 4y+18 + y<BR>\n" );
document.write( "128 = 5y+18<BR>\n" );
document.write( "5y+18 = 128<BR>\n" );
document.write( "5y = 128-18<BR>\n" );
document.write( "5y = 110<BR>\n" );
document.write( "y = 110/5<BR>\n" );
document.write( "y = 22\r<BR>\n" );
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document.write( "Let's find x based on that.<BR>\n" );
document.write( "x = 2y+9<BR>\n" );
document.write( "x = 2*22+9<BR>\n" );
document.write( "x = 44+9<BR>\n" );
document.write( "<font color=red>x = 53</font>\r<BR>\n" );
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document.write( "Since x = 53 and y = 22, we can then say: <font color=red>u5 = 53</font> and u4 = 22\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "u3 = 9<BR>\n" );
document.write( "u4 = 22<BR>\n" );
document.write( "<font color=red>u5</font> = 2*u4+u3 = 2*22+9 = 44+9 = <font color=red>53</font><BR>\n" );
document.write( "u6 = 2*u5+u4 = 2*53+22 = 106+22 = 128<BR>\n" );
document.write( "The answer is fully confirmed.\r<BR>\n" );
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document.write( "Again this all hinges on the assumption that the sequence is u_(n+2) = 2u_(n+1) + u_(n)<BR>\n" );
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