Problem 5.4 (Grade a "Proof"). Study the following claim as well as the "proof": Claim. For all n e N, 3| (n³ +44n). “Proof". When n = 1, n³+44n = 45 is divisible by 3. When n = 2, n³ +44n = 96 is divisible by 3. When n = 3, n³ + 44n = 159 is divisible by 3. When n = n3 + 44n = 1004400 is divisible by 3. Thus 3 | (n³ +44n) for all n E N. = 100, Complete the following questions concerning the above claim and "proof": (1) Determine whether the "proof" is rigorous. Identify the issues in the "proof", if any. (2) Determine whether the claim is true or false. Justify the answer in part (3). (3) If the the claim is true and the “proof" is not rigorous, then provide a correct and rigorous proof. If the claim is false, give a concrete counterexample. Hint. Complete the parts as instructed. If the claim is false, then (obviously) there is no way the "proof" could be correct.
Problem 5.4 (Grade a "Proof"). Study the following claim as well as the "proof": Claim. For all n e N, 3| (n³ +44n). “Proof". When n = 1, n³+44n = 45 is divisible by 3. When n = 2, n³ +44n = 96 is divisible by 3. When n = 3, n³ + 44n = 159 is divisible by 3. When n = n3 + 44n = 1004400 is divisible by 3. Thus 3 | (n³ +44n) for all n E N. = 100, Complete the following questions concerning the above claim and "proof": (1) Determine whether the "proof" is rigorous. Identify the issues in the "proof", if any. (2) Determine whether the claim is true or false. Justify the answer in part (3). (3) If the the claim is true and the “proof" is not rigorous, then provide a correct and rigorous proof. If the claim is false, give a concrete counterexample. Hint. Complete the parts as instructed. If the claim is false, then (obviously) there is no way the "proof" could be correct.
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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Transcribed Image Text:Problem 5.4 (Grade a "Proof"). Study the following claim as well as the "proof":
Claim. For all n e N, 3| (n³ +44n).
“Proof". When n = 1, n³ +44n = 45 is divisible by 3. When n = 2, n³ +44n = 96
is divisible by 3. When n = 3, n³ + 44n = 159 is divisible by 3. When n = 100,
n3 + 44n = 1004400 is divisible by 3. Thus 3 | (n³ +44n) for all n e N.
%3D
Complete the following questions concerning the above claim and "proof":
(1) Determine whether the "proof" is rigorous. Identify the issues in the "proof",
if any.
(2) Determine whether the claim is true or false. Justify the answer in part (3).
(3) If the the claim is true and the “proof" is not rigorous, then provide a correct and
rigorous proof. If the claim is false, give a concrete counterexample.
Hint. Complete the parts as instructed. If the claim is false, then (obviously) there is no
way the "proof" could be correct.
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