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Problem 2: Define a sequence rn where ro = 9, rị = 12, r2 = –6, and for integers n > 3, r, = 7rn–3. Prove that for all nonnegative integers n, 3|rn (tha is, each term in the sequence is divisible by 3).

Problem 2: Define a sequence rn where ro = 9, rị = 12, r2 = –6, and for integers n > 3, r, = 7rn–3. Prove that for all nonnegative integers n, 3|rn (tha is, each term in the sequence is divisible by 3).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove by induction

**Problem 2:** Define a sequence \( r_n \) where \( r_0 = 9 \), \( r_1 = 12 \), \( r_2 = -6 \), and for integers \( n \geq 3 \), \( r_n = 7r_{n-3} \). Prove that for all nonnegative integers \( n \), \( 3 \mid r_n \) (that is, each term in the sequence is divisible by 3). --- In this problem, you are given a recursive sequence and asked to prove a divisibility condition—specifically, that each term of the sequence is divisible by 3. The sequence starts with the initial terms \( r_0 = 9 \), \( r_1 = 12 \), \( r_2 = -6 \), and is defined recursively for \( n \geq 3 \) by \( r_n = 7r_{n-3} \). The task is to show that each \( r_n \) is divisible by 3 for all nonnegative integers \( n \). --- **Discussion:** To solve this problem, consider proving by induction or direct calculation, verifying that each term generated from the recursive relation remains divisible by 3. The initial terms are clearly divisible by 3. For \( n \geq 3 \), use the recursive formula and induction to prove the divisibility of each subsequent term.
Transcribed Image Text:**Problem 2:** Define a sequence \( r_n \) where \( r_0 = 9 \), \( r_1 = 12 \), \( r_2 = -6 \), and for integers \( n \geq 3 \), \( r_n = 7r_{n-3} \). Prove that for all nonnegative integers \( n \), \( 3 \mid r_n \) (that is, each term in the sequence is divisible by 3). --- In this problem, you are given a recursive sequence and asked to prove a divisibility condition—specifically, that each term of the sequence is divisible by 3. The sequence starts with the initial terms \( r_0 = 9 \), \( r_1 = 12 \), \( r_2 = -6 \), and is defined recursively for \( n \geq 3 \) by \( r_n = 7r_{n-3} \). The task is to show that each \( r_n \) is divisible by 3 for all nonnegative integers \( n \). --- **Discussion:** To solve this problem, consider proving by induction or direct calculation, verifying that each term generated from the recursive relation remains divisible by 3. The initial terms are clearly divisible by 3. For \( n \geq 3 \), use the recursive formula and induction to prove the divisibility of each subsequent term.
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