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Chapter 5: Problem 15

Solve each LHRRWCC. $$a_{n}=a_{n-1}+a_{n-2}, a_{0}=2, a_{1}=3$$

Short Answer

Expert verified
The general formula for the given LHRRWCC is: \[a_n = \frac{1 + \sqrt{5}}{2} \left(\frac{1 + \sqrt{5}}{2}\right)^n + \frac{1 - \sqrt{5}}{2} \left(\frac{1 - \sqrt{5}}{2}\right)^n\]

Step by step solution

01

Identify the characteristic equation from the given recurrence relation

Transform the given recurrence relation, $$a_{n} = a_{n-1} + a_{n-2}$$, into a characteristic equation by replacing \(a_n\) with \(r^n\), \(a_{n-1}\) with \(r^{n-1}\), and \(a_{n-2}\) with \(r^{n-2}\). The characteristic equation becomes: \(r^n = r^{n-1} + r^{n-2}\).
02

Simplify the characteristic equation

Factor out the term with the least exponent, which is \(r^{n-2}\), from the characteristic equation: \[r^{n-2}(r^2 - r - 1) = 0\]
03

Solve for r in the simplified characteristic equation

The roots of the equation will be the solutions for r. As the quadratic equation \(r^2 - r - 1 = 0\) doesn't have rational roots, we can use the quadratic formula to solve for r: \[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] So, \[r = \frac{1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)} = \frac{1 \pm \sqrt{5}}{2}\] Therefore, roots r1 and r2 are: $$r_1 = \frac{1 + \sqrt{5}}{2}$$ and $$r_2 = \frac{1 - \sqrt{5}}{2}$$
04

Use the general formula for LHRRWCC

The closed-form expression for the given LHRRWCC is: \[a_n = A(r_1)^n + B(r_2)^n\]
05

Use initial values to find A and B

Use provided initial values $$a_{0} = 2$$ and $$a_{1} = 3$$, and our expressions concerning roots $$r_1$$ and $$r_2$$ to create a system of equations: $$a_0 = A(r_1)^0 + B(r_2)^0 = 2$$ $$a_1 = A(r_1)^1 + B(r_2)^1 = 3$$ The system of equations becomes: \[A + B = 2\] \[A\frac{1 + \sqrt{5}}{2} + B\frac{1 - \sqrt{5}}{2} = 3\]
06

Solve the system of equations

Solve the system of equations for A and B using substitution or elimination method. After solving the system of equations, we get: $$A = \frac{1 + \sqrt{5}}{2}$$ and $$B = \frac{1 - \sqrt{5}}{2}$$
07

Write the final closed-form expression

Using the values of A and B, the general formula for the given LHRRWCC is: \[a_n = \frac{1 + \sqrt{5}}{2} \left(\frac{1 + \sqrt{5}}{2}\right)^n + \frac{1 - \sqrt{5}}{2} \left(\frac{1 - \sqrt{5}}{2}\right)^n\]

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Most popular questions from this chapter

Algorithm 5.10 computes the \(n\)th power of a positive real number \(x,\) where \(n \geq 0 .\) Use it to answer Exercises 18-24 . Algorithm exponentiation(x,n) (* This algorithm computes the nth power of \(x\) using recursion and returns the value in the variable answer.*) 0\. Begin (* algorithm *) 1\. if \(n=0\) then 2\. answer \(\leftarrow 1\) 3\. else 1 if \(n=1\) then 4\. answer \(\leftarrow x\) 5\. else 6\. begin (* else *) 7\. value \(\leftarrow\) exponentiation \((x,\lfloor n / 2\rfloor)\) 8\. answer \(\leftarrow\) value \(\cdot\) value 9\. If \(n\) is odd then 10\. answer \(\leftarrow\) answer \(\cdot\) \(x\) 11\. endelse 12\. End (* algorithm *) Let \(a_{n}\) denote the number of multiplications (lines \(7-10 )\) required by the algorithm to compute \(x^{n}\) . Compute each. $$a_{4}$$

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