5. (i) Define a sequence of numbers Rn, n = 1, 2, ... by setting R₁ = 7, R₂ = 21, and Rn+1 = 5Rn+ Rn−1, for n ≥ 2. So the sequence looks like Prove that 7,21, 112,518, 3017,... gcd(Rn+1, Rn) = 7. (ii) Define an analogous sequence Sn with gcd(Sn+1, Sn) = 23 and justify your answer.
5. (i) Define a sequence of numbers Rn, n = 1, 2, ... by setting R₁ = 7, R₂ = 21, and Rn+1 = 5Rn+ Rn−1, for n ≥ 2. So the sequence looks like Prove that 7,21, 112,518, 3017,... gcd(Rn+1, Rn) = 7. (ii) Define an analogous sequence Sn with gcd(Sn+1, Sn) = 23 and justify your answer.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...
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