Suppose that the sequences s0,s1,s2,... and t0, t1, t2,... both satisfy the same

Chapter 5, Problem 18

(choose chapter or problem)

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)