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Consider the third order difference equation An+3 = An+2 + an+1+an where ao = aj = 0 and a2 = 1. (This is often referred to as the "Tribonacci" Sequence.) Determine a.

Consider the third order difference equation An+3 = An+2 + an+1+an where ao = aj = 0 and a2 = 1. (This is often referred to as the "Tribonacci" Sequence.) Determine a.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Mathematics Topic: Tribonacci Sequence** **Consider the third order difference equation** \[ a_{n+3} = a_{n+2} + a_{n+1} + a_n \] **where** \[ a_0 = a_1 = 0 \quad \text{and} \quad a_2 = 1. \] (This is often referred to as the "Tribonacci" Sequence.) **Objective:** Determine \( a_9 \). **Explanation:** The Tribonacci sequence is similar to the Fibonacci sequence, but each term is the sum of the preceding three terms. Given the initial conditions \( a_0 = 0 \), \( a_1 = 0 \), and \( a_2 = 1 \), the sequence begins to develop as follows: - \( a_3 = a_2 + a_1 + a_0 = 1 + 0 + 0 = 1 \) - \( a_4 = a_3 + a_2 + a_1 = 1 + 1 + 0 = 2 \) - \( a_5 = a_4 + a_3 + a_2 = 2 + 1 + 1 = 4 \) - \( a_6 = a_5 + a_4 + a_3 = 4 + 2 + 1 = 7 \) - \( a_7 = a_6 + a_5 + a_4 = 7 + 4 + 2 = 13 \) - \( a_8 = a_7 + a_6 + a_5 = 13 + 7 + 4 = 24 \) - \( a_9 = a_8 + a_7 + a_6 = 24 + 13 + 7 = 44 \) Therefore, the ninth term, \( a_9 \), of the Tribonacci sequence is 44.
Transcribed Image Text:**Mathematics Topic: Tribonacci Sequence** **Consider the third order difference equation** \[ a_{n+3} = a_{n+2} + a_{n+1} + a_n \] **where** \[ a_0 = a_1 = 0 \quad \text{and} \quad a_2 = 1. \] (This is often referred to as the "Tribonacci" Sequence.) **Objective:** Determine \( a_9 \). **Explanation:** The Tribonacci sequence is similar to the Fibonacci sequence, but each term is the sum of the preceding three terms. Given the initial conditions \( a_0 = 0 \), \( a_1 = 0 \), and \( a_2 = 1 \), the sequence begins to develop as follows: - \( a_3 = a_2 + a_1 + a_0 = 1 + 0 + 0 = 1 \) - \( a_4 = a_3 + a_2 + a_1 = 1 + 1 + 0 = 2 \) - \( a_5 = a_4 + a_3 + a_2 = 2 + 1 + 1 = 4 \) - \( a_6 = a_5 + a_4 + a_3 = 4 + 2 + 1 = 7 \) - \( a_7 = a_6 + a_5 + a_4 = 7 + 4 + 2 = 13 \) - \( a_8 = a_7 + a_6 + a_5 = 13 + 7 + 4 = 24 \) - \( a_9 = a_8 + a_7 + a_6 = 24 + 13 + 7 = 44 \) Therefore, the ninth term, \( a_9 \), of the Tribonacci sequence is 44.
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