document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209830>Question 1209830</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let<BR>\n" );
document.write( "A_0 = 0<BR>\n" );
document.write( "A_1 = 1<BR>\n" );
document.write( "A_n = A_{n - 1} + A_{n - 2} for n ge 2\r<BR>\n" );
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document.write( "There is a unique ordered pair (c,d) such that c \alpha^n + d \beta^n is the closed form for sequence A_n.  Find the ordered pair (c,d).<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 #850736 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=850736\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You've described the Fibonacci sequence, where:\r<BR>\n" );
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document.write( "* $A_0 = 0$<BR>\n" );
document.write( "* $A_1 = 1$<BR>\n" );
document.write( "* $A_n = A_{n-1} + A_{n-2}$ for $n \ge 2$\r<BR>\n" );
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document.write( "We're looking for the closed-form expression of $A_n$ in the form $c \alpha^n + d \beta^n$, where $\alpha$ and $\beta$ are the roots of the characteristic equation.\r<BR>\n" );
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document.write( "**1. Find the Characteristic Equation and its Roots**\r<BR>\n" );
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document.write( "The characteristic equation for the recurrence relation is:\r<BR>\n" );
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document.write( "* $x^2 - x - 1 = 0$\r<BR>\n" );
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document.write( "Using the quadratic formula, the roots are:\r<BR>\n" );
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document.write( "* $\alpha = \frac{1 + \sqrt{5}}{2}$ (the golden ratio)<BR>\n" );
document.write( "* $\beta = \frac{1 - \sqrt{5}}{2}$\r<BR>\n" );
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document.write( "**2. Use Initial Conditions to Solve for c and d**\r<BR>\n" );
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document.write( "We have the following initial conditions:\r<BR>\n" );
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document.write( "* $A_0 = 0$<BR>\n" );
document.write( "* $A_1 = 1$\r<BR>\n" );
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document.write( "Substitute these into the closed-form expression:\r<BR>\n" );
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document.write( "* For $n = 0$: $A_0 = c \alpha^0 + d \beta^0 = c + d = 0$<BR>\n" );
document.write( "* For $n = 1$: $A_1 = c \alpha^1 + d \beta^1 = c \left( \frac{1 + \sqrt{5}}{2} \right) + d \left( \frac{1 - \sqrt{5}}{2} \right) = 1$\r<BR>\n" );
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document.write( "From the first equation, we get $d = -c$. Substitute this into the second equation:\r<BR>\n" );
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document.write( "* $c \left( \frac{1 + \sqrt{5}}{2} \right) - c \left( \frac{1 - \sqrt{5}}{2} \right) = 1$<BR>\n" );
document.write( "* $c \left( \frac{1 + \sqrt{5} - (1 - \sqrt{5})}{2} \right) = 1$<BR>\n" );
document.write( "* $c \left( \frac{2 \sqrt{5}}{2} \right) = 1$<BR>\n" );
document.write( "* $c \sqrt{5} = 1$<BR>\n" );
document.write( "* $c = \frac{1}{\sqrt{5}}$\r<BR>\n" );
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document.write( "Now, substitute $c$ back into $d = -c$:\r<BR>\n" );
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document.write( "* $d = -\frac{1}{\sqrt{5}}$\r<BR>\n" );
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document.write( "**3. The Ordered Pair (c, d)**\r<BR>\n" );
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document.write( "Therefore, the ordered pair $(c, d)$ is:\r<BR>\n" );
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document.write( "* $\left( \frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}} \right)$\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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