1. Consider the sequence defined by a; = (-1)* · 2* for i > 0. Compute: (a) i%3D0 (b) 2. Prove or disprove: For all r € Z, if æ mod 7 = 5, then r² mod 7 = 4.

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A: We have to find given sequence is convergent or not .

Q: a) ɑn Зп+1

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

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Q: 2n b) ɑn n²+1

A: b) To find the first five terms of the given sequence is: an=2nn2+1

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A: We prove this by strong Mathematical induction.

Q: 1. Consider the sequence defined by a = (-1)' · 2ª for i 2 0. Compute: (a) (b) I, azi

A: ∑i=02a2i=∑i=02-12i22i            =∑i=024i            =40+4+42            =21

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Q: - Determine the Collatz sequence for Co =7.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Consider the following sequence defined recursively: a1 %3D 5, аz 3D 8, аm+1 —D 2а, — ап-: 2an - an-

A: Introduction: A recurrence relation is a rule-based equation that represents a sequence. It aids in…

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A: We use the Cauchy criterion to prove the result.

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A: Calculus   sequence

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A: Since you have asked multiple questions, we will solve first question for you. If you want other…

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A: I have proved using Apsilon delta definition

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Q: (E) The sequence defined by a, =1, a=-a, converges to 1. e

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