Define a sequence a1, a2, a3,... as follows: a1 = 1, a2 = 3, and ak = ak1 + ak2 for all

Chapter 5, Problem 9

(choose chapter or problem)

Define a sequence \(a_{1}, a_{2}, a_{3}\), . . . as follows: \(a_{1} = 1, a_{2} = 3\), and \(a_{k} = a_{k−1} + a_{k−2}\) for all integers k ≥ 3. (This sequence is known as the Lucas sequence.) Use strong mathematical induction to prove that \(a_{n} ≤ \frac {7}{4}^{n}\) for all integers n ≥ 1.

Text Transcription:

a_1, a_2, a_3

a_1 = 1, a_2 = 3

a_k = a_k−1 + a_k−2

a_n ≤ 7 / 4^n