document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209828>Question 1209828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Fibonacci sequence, is defined by F_0 = 0, F_1 = 1, and F_n = F_{n - 2} + F_{n - 1}.  It turns out that<BR>\n" );
document.write( "F_n = \frac{\alpha^n - \beta^n}{\sqrt{5}},<BR>\n" );
document.write( "where \alpha = \frac{1 + \sqrt{5}}{2} and \beta = \frac{1 - \sqrt{5}}{2}.\r<BR>\n" );
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document.write( "The Lucas sequence is defined as follows: L_0 = 2, L_1 = 1, and<BR>\n" );
document.write( "L_n = L_{n - 1} + L_{n - 2}<BR>\n" );
document.write( "for n \ge 2.  What is L_4?<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 #850743 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850743\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The sequence with first two terms 2 and 1 with the recursive definition that each term is the sum of the previous two terms is NOT \"THE\" Lucas sequence.  A Lucas sequence is ANY sequence in which each term is a linear combination of the two preceding terms (and the first two terms can be any numbers).<br><BR>\n" );
document.write( "The sequence in this problem is A Lucas sequence with first two terms 2 and 1.<br><BR>\n" );
document.write( "Subsequent terms of the sequence are found using the given recursive definition.<br><BR>\n" );
document.write( "L(0)=2<BR>\n" );
document.write( "L(1)=1<BR>\n" );
document.write( "L(2)=L(0)+L(1)=2+1=3<BR>\n" );
document.write( "L(3)=L(1)+L(2)=1+3=4<BR>\n" );
document.write( "L(4)=L(2)+L(3)=3+4=7<BR>\n" );
document.write( "L(5)=L(3)+L(4)+4+7=11<BR>\n" );
document.write( "etc...<br><BR>\n" );
document.write( "ANSWER: L(4)=7<br><BR>\n" );
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