document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167599>Question 1167599</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an explicit formula in terms of n for a sequence <BR>\n" );
document.write( "b0, b1, b2,   <BR>\n" );
document.write( " that satisfies<BR>\n" );
document.write( "bk = 2b_k − 1 − 10b_k − 2 for each integer k ≥ 2<BR>\n" );
document.write( "with initial conditions <BR>\n" );
document.write( "b0 = 5<BR>\n" );
document.write( " and <BR>\n" );
document.write( "b1 = 5.<BR>\n" );
document.write( " (Your answer will involve complex numbers.)<BR>\n" );
document.write( "bn =     for every integer n ≥ 0\r<BR>\n" );
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document.write( "Discrete math homework question<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 #852321 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=852321\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem requires solving a second-order homogeneous linear recurrence relation with constant coefficients.\r<BR>\n" );
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document.write( "## Recurrence Relation and Initial Conditions\r<BR>\n" );
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document.write( "The recurrence relation is:<BR>\n" );
document.write( "$$b_k = 2b_{k-1} - 10b_{k-2} \quad \text{for } k \ge 2$$<BR>\n" );
document.write( "The initial conditions are:<BR>\n" );
document.write( "$$b_0 = 5$$<BR>\n" );
document.write( "$$b_1 = 5$$\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## 1. The Characteristic Equation\r<BR>\n" );
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document.write( "The characteristic equation is derived by assuming a solution of the form $b_n = r^n$:<BR>\n" );
document.write( "$$r^2 = 2r - 10$$<BR>\n" );
document.write( "$$r^2 - 2r + 10 = 0$$\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## 2. Finding the Roots\r<BR>\n" );
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document.write( "We use the quadratic formula to find the roots $r$:<BR>\n" );
document.write( "$$r = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(10)}}{2(1)}$$<BR>\n" );
document.write( "$$r = \frac{2 \pm \sqrt{4 - 40}}{2}$$<BR>\n" );
document.write( "$$r = \frac{2 \pm \sqrt{-36}}{2}$$<BR>\n" );
document.write( "$$r = \frac{2 \pm 6i}{2}$$<BR>\n" );
document.write( "$$r_1 = 1 + 3i \quad \text{and} \quad r_2 = 1 - 3i$$\r<BR>\n" );
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document.write( "Since the roots are complex, we convert them to polar form, $r = \rho e^{i\phi}$:<BR>\n" );
document.write( "$$\rho = |r_1| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}$$<BR>\n" );
document.write( "$$\phi = \arctan\left(\frac{3}{1}\right) = \arctan(3)$$\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## 3. General Solution Form\r<BR>\n" );
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document.write( "The general solution for a recurrence with distinct complex roots $r_1$ and $r_2$ is:<BR>\n" );
document.write( "$$b_n = c_1 r_1^n + c_2 r_2^n$$<BR>\n" );
document.write( "Alternatively, using the polar form:<BR>\n" );
document.write( "$$b_n = \rho^n (A \cos(n\phi) + B \sin(n\phi))$$<BR>\n" );
document.write( "Using the roots $r_1 = 1 + 3i$ and $r_2 = 1 - 3i$, the general solution is:<BR>\n" );
document.write( "$$b_n = c_1(1 + 3i)^n + c_2(1 - 3i)^n$$\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## 4. Applying Initial Conditions\r<BR>\n" );
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document.write( "We use the initial conditions $b_0 = 5$ and $b_1 = 5$ to solve for $c_1$ and $c_2$.\r<BR>\n" );
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document.write( "**For $n=0$:**<BR>\n" );
document.write( "$$b_0 = c_1(1 + 3i)^0 + c_2(1 - 3i)^0$$<BR>\n" );
document.write( "$$5 = c_1(1) + c_2(1)$$<BR>\n" );
document.write( "$$\mathbf{c_1 + c_2 = 5} \quad \text{(Equation 1)}$$\r<BR>\n" );
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document.write( "**For $n=1$:**<BR>\n" );
document.write( "$$b_1 = c_1(1 + 3i)^1 + c_2(1 - 3i)^1$$<BR>\n" );
document.write( "$$5 = c_1(1 + 3i) + c_2(1 - 3i)$$<BR>\n" );
document.write( "$$5 = c_1 + 3ic_1 + c_2 - 3ic_2$$<BR>\n" );
document.write( "$$5 = (c_1 + c_2) + 3i(c_1 - c_2)$$\r<BR>\n" );
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document.write( "Substitute $c_1 + c_2 = 5$ from Equation 1:<BR>\n" );
document.write( "$$5 = 5 + 3i(c_1 - c_2)$$<BR>\n" );
document.write( "$$0 = 3i(c_1 - c_2)$$<BR>\n" );
document.write( "Since $3i \neq 0$, we must have:<BR>\n" );
document.write( "$$c_1 - c_2 = 0$$<BR>\n" );
document.write( "$$\mathbf{c_1 = c_2} \quad \text{(Equation 2)}$$\r<BR>\n" );
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document.write( "Substitute Equation 2 into Equation 1:<BR>\n" );
document.write( "$$c_1 + c_1 = 5$$<BR>\n" );
document.write( "$$2c_1 = 5$$<BR>\n" );
document.write( "$$\mathbf{c_1 = \frac{5}{2}}$$<BR>\n" );
document.write( "Therefore, $\mathbf{c_2 = \frac{5}{2}}$.\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## 5. Explicit Formula\r<BR>\n" );
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document.write( "Substituting the values of $c_1$ and $c_2$ into the general solution:<BR>\n" );
document.write( "$$b_n = \frac{5}{2}(1 + 3i)^n + \frac{5}{2}(1 - 3i)^n$$<BR>\n" );
document.write( "$$b_n = \frac{5}{2} \left[ (1 + 3i)^n + (1 - 3i)^n \right]$$\r<BR>\n" );
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document.write( "This is the explicit formula.\r<BR>\n" );
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document.write( "$$b_n = \mathbf{\frac{5}{2} (1 + 3i)^n + \frac{5}{2} (1 - 3i)^n}$$\r<BR>\n" );
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document.write( "*(Note: While the formula involves complex numbers, the result for any integer $n \ge 0$ will always be a real number. This is because $b_n$ is the sum of a complex number and its conjugate, which always yields $2 \times \text{Re}(\text{complex number})$.)* </SPAN>\n" );
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