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4. Let P(n) be the predicate "n2 – n is even". Prove that P(n) → P(n + 1). (This problem is slightly challenging.)

4. Let P(n) be the predicate "n2 – n is even". Prove that P(n) → P(n + 1). (This problem is slightly challenging.)

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Hi! This is a question on Discrete Math/Structures 

**Problem 4: Proving Mathematical Induction** **Objective**: Learn how to tackle a slightly challenging problem using mathematical induction. --- **Problem Statement**: Let \( P(n) \) be the predicate " \( n^2 - n \) is even". Prove that \( P(n) \rightarrow P(n+1) \). (This problem is slightly challenging.) --- **The Approach**: 1. **Understand the Predicate**: - \( P(n) \) means that \( n^2 - n \) is an even number. - We will explore \( P(n+1) \) means \( (n+1)^2 - (n+1) \) is an even number. 2. **Induction Hypothesis**: - Assume that \( n^2 - n \) is even for some integer \( n \). 3. **Prove the Inductive Step**: - Show that if \( n^2 - n \) is even, then \((n+1)^2 - (n+1)\) is also even. 4. **Verification**: - Substitute and check if \( (n+1)^2 - (n+1) \equiv n^2 + 2n + 1 - n - 1 \). - Simplify to verify if the resulting expression is even. --- This exercise not only strengthens the understanding of mathematical induction but also enhances problem-solving skills in algebraic expressions. Be sure to check your steps and verify your solution.
Transcribed Image Text:**Problem 4: Proving Mathematical Induction** **Objective**: Learn how to tackle a slightly challenging problem using mathematical induction. --- **Problem Statement**: Let \( P(n) \) be the predicate " \( n^2 - n \) is even". Prove that \( P(n) \rightarrow P(n+1) \). (This problem is slightly challenging.) --- **The Approach**: 1. **Understand the Predicate**: - \( P(n) \) means that \( n^2 - n \) is an even number. - We will explore \( P(n+1) \) means \( (n+1)^2 - (n+1) \) is an even number. 2. **Induction Hypothesis**: - Assume that \( n^2 - n \) is even for some integer \( n \). 3. **Prove the Inductive Step**: - Show that if \( n^2 - n \) is even, then \((n+1)^2 - (n+1)\) is also even. 4. **Verification**: - Substitute and check if \( (n+1)^2 - (n+1) \equiv n^2 + 2n + 1 - n - 1 \). - Simplify to verify if the resulting expression is even. --- This exercise not only strengthens the understanding of mathematical induction but also enhances problem-solving skills in algebraic expressions. Be sure to check your steps and verify your solution.
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