Suppose that h0, h1, h2,... is a sequence defined as follows: h0 = 1, h1 = 2, h2 = 3, hk

Chapter 5, Problem 8

(choose chapter or problem)

Suppose that \(h_{0}, h_{1}, h_{2}\), . . . is a sequence defined as follows:

\(h_{0} = 1, h_{1} = 2, h_{2} = 3\),

\(h_{k} = h_{k−1} + h_{k−2} + h_{k−3}\) for all integers k ≥ 3.

a. Prove that \(h_{n} ≤ 3^{n}\) for all integers n ≥ 0.

b. Suppose that s is any real number such that \(s^{3} ≥ s^{2} + s + 1\). (This implies that s > 1.83.) Prove that \(h_{n} ≤ s^{n}\) for all n ≥ 2.

Text Transcription:

h_0, h_1, h_2

h_0 = 1, h_1 = 2, h_2 = 3

h_k = h_k−1 + h_k−2 + h_k−3

h_n ≤ 3^n

s^3 ≥ s^2 + s + 1

h_n ≤ s^n