Show that if r is a nonzero real number, k and m are distinct integers, and ak and am

Which of the following are second-order linear homogeneous recurrence relations with constant coefficients? a. \(a_{k} = 2a_{k?1} ? 5a_{k?2}\) b. \(b_{k} = kb_{k?1} + b_{k?2}\) c. \(c_{k} = 3c_{k?1} \cdot c^{2}_{k?2}\) d. \(d_{k} = 3d_{k?1} + d_{k?2}\) e. \(r_{k} = r_{k?1} ? r_{k?2} ?2\) f.\(s_{k} = 10s_{k?2}\) Text Transcription: a_k = 2a_k?1 ? 5a_k?2 b_k = kb_k?1 + b_k?2 c_k = 3c_k?1 cdot c^2_k?2 d_k = 3d_k?1 + d_k?2 r_k = r_k?1 ? r_k?2 ?2 s_k = 10s_k?2

Which of the following are second-order linear homogeneous recurrence relations with constant coefficients? a. \(a_{k} = (k ? 1)a_{k?1} + 2ka_{k?2}\) b. \(b_{k} = ?b_{k?1} + 7b_{k?2}\) c. \(c_{k} = 3c_{k?1} + 1\) d. \(d_{k} = 3d^{2}_{k?1}+ d_{k?2}\) e. \(r_{k} = r_{k?1} ? 6r_{k?3}\) f.\(s_{k} = s_{k?1} + 10s_{k?2}\) Text Transcription: a_k = (k ? 1)a_k?1 + 2ka_k?2 b_k = ?b_k?1 + 7b_k?2 c_k = 3c_k?1 + 1 d_k = 3d^2_k?1+ d_k?2 r_k = r_k?1 ? 6r_k?3 s_k = s_k?1 + 10s_k?2

Let \(a_{0}, a_{1}, a_{2}\), . . . be the sequence defined by the explicit formula \(a_{n} = C \cdot 2^{n} + D\) for all integers n ? 0, where C and D are real numbers. a. Find C and D so that \(a_{0} = 1\) and \(a_{1} = 3\). What is a2 in this case? b. Find C and D so that \(a_{0} = 0\) and \(a_{1} = 2\). What is a2 in this case? Text Transcription: a_0, a_1, a_2 a_n = C cdot 2^n + D a_0 = 1 a_1 = 3 a_0 = 0 a_1 = 2

Let \(b_{0}, b_{1}, b_{2}\), . . . be the sequence defined by the explicit formula \(b_{n} = C \cdot 3^{n} + D(?2)^{n}\) for all integers n ? 0, where C and D are real numbers. a. Find C and D so that \(b_{0} = 0\) and \(b_{1} = 5\). What is \(b_{2}\) in this case? b. Find C and D so that \(b_{0} = 3\) and \(b_{1} = 4\). What is \(b_{2}\) in this case? Text Transcription: b_0, b_1, b_2 b_n = C cdot 3^n + D(?2)^n b_0 = 0 b_1 = 5 b_2 b_0 = 3 b_1 = 4

Let \(a_{0}, a_{1}, a_{2}\), . . . be the sequence defined by the explicit formula \(a_{n} = C \cdot 2^{n} + D\) for all integers n ? 0, where C and D are real numbers. Show that for any choice of C and D, \(a_{k} = 3a_{k?1} ? 2a_{k?2}\) for all integers k ? 2. Text Transcription: a_0, a_1, a_2 a_n = C cdot 2^n + D a_k = 3a_k?1 ? 2a_k?2

Let \(b_{0}, b_{1}, b_{2}\), . . . be the sequence defined by the explicit formula \(b_{n} = C \cdot 3^{n} + D(?2)^{n}\) for all integers n ? 0, where C and D are real numbers. Show that for any choice of C and D, \(b_{k} = b_{k?1} + 6b_{k?2}\) for all integers k ? 2. Text Transcription: b_0, b_1, b_2 b_n = C cdot 3^n + D(?2)^n b_k = b_k?1 + 6b_k?2

Solve the system of equations in Example 5.8.4 to obtain \(C=\frac{1+\sqrt{5}}{2 \sqrt{5}} \text { and } D=\frac{-(1-\sqrt{5})}{2 \sqrt{5}}\). Text Transcription: C = 1 + sqrt 5 / 2 sqrt 5 and D = -(1 - sqrt 5) / 2 sqrt 5

In each of 8–10: (a) suppose a sequence of the form \(1.t.t^_{2}.t^{3} . . . t^{n} . . .\) where t = 0, satisfies the given recurrence relation (but not necessarily the initial conditions), and find all possible values of t: (b) suppose a sequence satisfies the given initial conditions as well as the recurrence relation, and find an explicit formula for the sequence. \(a_{k} = 2a_{k?1} + 3a_{k?2}\), for all integers k ? 2 \(a_{0} = 1, a_{1} = 2\) Text Transcription: 1.t.t^2.t^3 . . . t^n . . . a_k = 2a_k?1 + 3a_k?2 a0 = 1, a1 = 2

In each of 8–10: (a) suppose a sequence of the form \(1.t.t^_{2}.t^{3} . . . t^{n} . . .\) where t = 0, satisfies the given recurrence relation (but not necessarily the initial conditions), and find all possible values of t: (b) suppose a sequence satisfies the given initial conditions as well as the recurrence relation, and find an explicit formula for the sequence. \(b_{k} = 7b_{k?1} ? 10b_{k?2}\), for all integers k ? 2 \(b_{0} = 2, b_{1} = 2\) Text Transcription: 1.t.t^2.t^3 . . . t^n . . . b_k = 7b_k?1 ? 10b_k?2 b_0 = 2, b_1 = 2

In each of 8–10: (a) suppose a sequence of the form \(1.t.t^_{2}.t^{3} . . . t^{n} . . .\) where t = 0, satisfies the given recurrence relation (but not necessarily the initial conditions), and find all possible values of t: (b) suppose a sequence satisfies the given initial conditions as well as the recurrence relation, and find an explicit formula for the sequence. \(c_{k} = c_{k?1} + 6c_{k?2},\), for all integers k ? 2 \(c_{0} = 0, c_{1} = 3\) Text Transcription: 1.t.t^2.t^3 . . . t^n . . . c_k = c_k?1 + 6c_k?2, c_0 = 0, c_1 = 3

In each of 11–16 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. \(d_{k} = 4d_{k?2}\), for all integers k ? 2 \(d_{0} = 1, d_{1} = ?1\) Text Transcription: d_k = 4d_k?2 d_0 = 1, d_1 = ?1

In each of 11–16 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. \(e_{k} = 9e_{k?2}\), for all integers k ? 2 \(e_{0} = 0, e_{1} = 2\) Text Transcription: e_k = 9e_k?2 e_0 = 0, e_1 = 2

In each of 11–16 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. \(r_{k} = 2r_{k?1} ? r_{k?2}\), for all integers k ? 2 \(r_{0} = 1, e_{1} = 4\) Text Transcription: r_k = 2r_k?1 ? r_k?2 r_0 = 1, r_1 = 4

In each of 11–16 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. \(s_{k} = ?4s_{k?1} ? 4s_{k?2}\), for all integers k ? 2 \(s_{0} = 4, s_{1} = -1\) Text Transcription: s_k = ?4s_k?1 ? 4s_k?2 s_0 = 4, s_1 = -1

In each of 11–16 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. \(t_{k} = 6t_{k?1} ? 9t_{k?2}\), for all integers k ? 2 \(t_{0} = 1, t_{1} = 3\) Text Transcription: t_k = 6t_k?1 ? 9t_k?2 t_0 = 1, t_1 = 3

In each of 11–16 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. \(s_{k} = 2s_{k?1} + 2s_{k?2}\), for all integers k ? 2 \(s_{0} = 1, s_{1} = 3\) Text Transcription: s_k = 2s_k?1 + 2s_k?2 s_0 = 1, s_1 = 3

Find an explicit formula for the sequence of exercise 39 in Section 5.6

Suppose that the sequences \(s_{0}, s_{1}, s_{2}, . . . and t_{0}, t_{1}, t_{2}, . . .\) both satisfy the same second-order linear homogeneous recurrence relation with constant coefficients: \(s_{k} = 5s_{k?1} ? 4s_{k?2}\) for all integers k ? 2, \(t_{k} = 5t_{k?1} ? 4t_{k?2}\) for all integers k ? 2. Show that the sequence \(2s_{0} + 3t_{0}, 2s_{1} + 3t_{1}, 2s_{2} + 3t_{2}, . . .\) also satisfies the same relation. In other words, show that \(2s_{k} + 3t_{k} = 5(2s_{k?1} + 3t_{k?1}) ? 4(2s_{k?2} + 3t_{k?2})\) for all integers k ? 2. Do not use Lemma 5.8.2. Text Transcription: s_0, s_1, s_2, . . . and t_0, t_1, t_2, . . . s_k = 5s_k?1 ? 4s_k?2 t_k = 5t_k?1 ? 4t_k?2 2s_0 + 3_t0, 2s_1 + 3t_1, 2s_2 + 3t_2, . . . 2s_k + 3t_k = 5(2s_k?1 + 3t_k?1) ? 4(2s_k?2 + 3t_k?2)

Show that if r, s, \(a_{0}\), and \(a_{1}\) are numbers with r = s, then there exist unique numbers C and D so that \(C + D = a_{0}\) \(Cr + Ds = a_{1}\). Text Transcription: a_0 a_1 C + D = a_0 Cr + Ds = a_1

Show that if r is a nonzero real number, k and m are distinct integers, and \(a_{k}\) and \(a_{m}\) are any real numbers, then there exist unique real numbers C and D so that \(Cr^{k} + kDr^{k} = a_{k}\) \(Cr^{m} + lDr^{m} = a_{m}\). Text Transcription: a_k a_m Cr^k + kDr^k = a_k Cr^m + lDr^m = a_m

Prove Theorem 5.8.5 for the case where the values of C and D are determined by \(a_{0}\) and \(a_{1}\). Text Transcription: a_0 a_1

Exercises 22 and 23 are intended for students who are familiar with complex numbers. Find an explicit formula for a sequence \(a_{0}, a_{1}, a_{2}, . . .\) that satisfies \(a_{k} = 2a_{k?1} ? 2a_{k?2}\) for all integers k ? 2 with initial conditions \(a_{0} = 1\) and \(a_{1} = 2\). Text Transcription: a_0, a_1, a_2, . . . a_k = 2a_k?1 ? 2a_k?2 a_0 = 1 a_1 = 2

Exercises 22 and 23 are intended for students who are familiar with complex numbers. Find an explicit formula for a sequence \(b_{0}, b_{1}, b_{2}, . . .\) that satisfies \(b_{k} = 2b_{k?1} ? 5b_{k?2}\) for all integers k ? 2 with initial conditions \(b_{0} = 1\) and \(b_{1} = 1\). Text Transcription: b_0, b_1, b_2, . . . b_k = 2b_k?1 ? 5b_k?2 b_0 = 1 b_1 = 1

The numbers \(\frac{1+\sqrt{5}}{2}\) and \(\frac{1-\sqrt{5}}{2}\) that appear in the explicit formula for the Fibonacci sequence are related to a quantity called the golden ratio in Greek mathematics. Consider a rectangle of length ? units and height 1, where ? > 1. Divide the rectangle into a rectangle and a square as shown in the preceding diagram. The square is 1 unit on each side, and the rectangle has sides of lengths 1 and ? ? 1. The ancient Greeks considered the outer rectangle to be perfectly proportioned (saying that the lengths of its sides were in a golden ratio to each other) if the ratio of the length to the width of the outer rectangle equaled the ratio of the length to the width of the inner rectangle. That is, \(\frac{\phi}{1}=\frac{1}{\phi-1}\). a. Show that ? satisfies the following quadratic equation: \(t^{2} ? t ? 1 = 0\). b. Find the two solutions of \(t^{2} ? t ? 1 = 0\) and call them ?1 and ?2. c. Express the explicit formula for the Fibonacci sequence in terms of ?1 and ?2. Text Transcription: 1 + sqrt 5 / 2 1 - sqrt 5 / 2 ? / 1 = 1 / ? ? 1 t^2 ? t ? 1 = 0