I am struggling with question 2 and 3. 1. Use induction to prove that, for all n € Z+,(i) 6 (n³ - n)(ii) 2+5+8+·+ (3n-1) = n(3n+1)/2=2. Suppose that an is defined by setting a₁ = -2, a₂ = 0, and an 4an-1-4an-2,where n ≥ 3. Using strong induction, prove that an = 2" (n − 2) for all n € Z+.-3. Let a be a positive integer. Evaluate the following:(i) ged(a, a²)(ii) ged(a, a²+1)(iii) ged(a, a+2) (Hint: consider two cases: when a is even and when a is odd)You do not need to use the Euclidean Algorithm to answer this question.

Answered: Suppose that an is defined by setting a…

Suppose that an is defined by setting a = = -2, a₂ = 0, and an = 4an-1-4an-29 where n ≥ 3. Using strong induction, prove that an = 2" (n-2) for all n € Z+.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 46E: Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection...

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Question

I am struggling with question 2 and 3.

1. Use induction to prove that, for all n € Z+,
(i) 6 (n³ - n)
(ii) 2+5+8+·
+ (3n-1) = n(3n+1)/2
=
2. Suppose that an is defined by setting a₁ = -2, a₂ = 0, and an 4an-1-4an-2,
where n ≥ 3. Using strong induction, prove that an = 2" (n − 2) for all n € Z+.
-
3. Let a be a positive integer. Evaluate the following:
(i) ged(a, a²)
(ii) ged(a, a²+1)
(iii) ged(a, a+2) (Hint: consider two cases: when a is even and when a is odd)
You do not need to use the Euclidean Algorithm to answer this question.

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