document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209829>Question 1209829</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the ordered pair (p,q) such that<BR>\n" );
document.write( "F_n = p \alpha^n + q \beta^n. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850735 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=850735\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $F_n$ denote the $n$-th Fibonacci number, where $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for $n \ge 0$. The characteristic equation for this recurrence is $x^2 - x - 1 = 0$. The roots of this equation are<BR>\n" );
document.write( "\[ \alpha = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad \beta = \frac{1 - \sqrt{5}}{2}. \]<BR>\n" );
document.write( "We are given that $F_n = p \alpha^n + q \beta^n$. We need to find the values of $p$ and $q$.\r<BR>\n" );
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document.write( "We can use the initial values $F_0 = 0$ and $F_1 = 1$.<BR>\n" );
document.write( "For $n=0$, we have $F_0 = p \alpha^0 + q \beta^0 = p + q = 0$.<BR>\n" );
document.write( "For $n=1$, we have $F_1 = p \alpha^1 + q \beta^1 = p \alpha + q \beta = 1$.\r<BR>\n" );
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document.write( "From the first equation, $q = -p$. Substituting this into the second equation, we get<BR>\n" );
document.write( "\[ p \alpha - p \beta = 1 \]<BR>\n" );
document.write( "\[ p (\alpha - \beta) = 1 \]<BR>\n" );
document.write( "\[ p \left( \frac{1 + \sqrt{5}}{2} - \frac{1 - \sqrt{5}}{2} \right) = 1 \]<BR>\n" );
document.write( "\[ p \left( \frac{2 \sqrt{5}}{2} \right) = 1 \]<BR>\n" );
document.write( "\[ p \sqrt{5} = 1 \]<BR>\n" );
document.write( "\[ p = \frac{1}{\sqrt{5}}. \]<BR>\n" );
document.write( "Since $q = -p$, we have $q = -\frac{1}{\sqrt{5}}$.\r<BR>\n" );
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document.write( "Therefore, the ordered pair $(p,q)$ is<BR>\n" );
document.write( "\[ \left( \frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}} \right). \]\r<BR>\n" );
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document.write( "Thus, the formula for the $n$-th Fibonacci number is<BR>\n" );
document.write( "\[ F_n = \frac{1}{\sqrt{5}} \left( \frac{1 + \sqrt{5}}{2} \right)^n - \frac{1}{\sqrt{5}} \left( \frac{1 - \sqrt{5}}{2} \right)^n. \]\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\left( \frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}} \right)}$<BR>\n" );
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