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Problem 2: (a) Use mathematical induction to prove that 3- 5" >n. 3" + 4"+1 for all integers n > 3. Hint: Note that n > n+1 for sufficiently large integers n. (How large n has to be in order for this inequality to hold?) This inequality could be useful in the inductive step. (b) Let g(n) = n · 3" + 4"+1 and h(n) = 5". Using the inequality from part (a) prove that g(n) = 0(h(n)). You need to give a rigorous proof derived directly from the definition of the O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = O(h(n)) follows from this definition.)

Problem 2: (a) Use mathematical induction to prove that 3- 5" >n. 3" + 4"+1 for all integers n > 3. Hint: Note that n > n+1 for sufficiently large integers n. (How large n has to be in order for this inequality to hold?) This inequality could be useful in the inductive step. (b) Let g(n) = n · 3" + 4"+1 and h(n) = 5". Using the inequality from part (a) prove that g(n) = 0(h(n)). You need to give a rigorous proof derived directly from the definition of the O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = O(h(n)) follows from this definition.)

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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Problem 2: (a) Use mathematical induction to prove that 3. 5" >n· 3" + 4"+1 for all integers n > 3. Hint: Note that n > n+1 for sufficiently large integers n. (How large n has to be in order for this inequality to hold?) This inequality could be useful in the inductive step. (b) Let g(n) =n· 3" + 4"+1 and h(n) = 5". Using the inequality from part (a) prove that g(n) = 0(h(n)). You need to give a rigorous proof derived directly from the definition of the O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = 0(h(n)) follows from this definition.)
Transcribed Image Text:Problem 2: (a) Use mathematical induction to prove that 3. 5" >n· 3" + 4"+1 for all integers n > 3. Hint: Note that n > n+1 for sufficiently large integers n. (How large n has to be in order for this inequality to hold?) This inequality could be useful in the inductive step. (b) Let g(n) =n· 3" + 4"+1 and h(n) = 5". Using the inequality from part (a) prove that g(n) = 0(h(n)). You need to give a rigorous proof derived directly from the definition of the O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = 0(h(n)) follows from this definition.)
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