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Problem 2: (a) Use properties of quadratic functions to prove that 5x² > (x+1)² for all real x > 1. (b) Use mathematical induction and the inequality from part (a) to prove that 3 - 5" > 4n+1 + n² for all integers n > 2.

Problem 2: (a) Use properties of quadratic functions to prove that 5x² > (x+1)² for all real x > 1. (b) Use mathematical induction and the inequality from part (a) to prove that 3 - 5" > 4n+1 + n² for all integers n > 2.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.5: Iterative Methods For Solving Linear Systems
Problem 27EQ
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Problem 2: (a) Use properties of quadratic functions to prove that 5x? > (x + 1)² for all real x > 1. (b) Use mathematical induction and the inequality from part (a) to prove that 3 · 5" > 4"+1 + n² for all integers n > 2. (c) Let g(n) = 4"+1 + n² 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 properties of quadratic functions to prove that 5x? > (x + 1)² for all real x > 1. (b) Use mathematical induction and the inequality from part (a) to prove that 3 · 5" > 4"+1 + n² for all integers n > 2. (c) Let g(n) = 4"+1 + n² 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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