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1. Show that 1+2+3+4++ n = (n+¹) for integers n ≥ 1 using induction

1. Show that 1+2+3+4++ n = (n+¹) for integers n ≥ 1 using induction

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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**Learning Target R4 Core**: Given a statement to be proven by (weak) induction, I can state and prove the base case, state the inductive hypothesis, and outline the proof. I can describe the subtle difference between 'weak' and 'strong' induction. **Please remember** the academic integrity policy that you swore to uphold. Proofs can be challenging. Just try your best. If you don’t get it this week, you have many more opportunities! You might not be able to prove these claims with what we learned in this class, and that’s okay. You’re just setting it up! For each statement below, complete each of the following parts: - State and prove the base case - State the inductive hypothesis - Outline how the rest of the proof would go, that is, tell me roughly what will happen to complete the proof including the conclusion you will arrive at; you do not need to actually do the complete proof. Remember if you have a question about this, you can email/send me a message on Teams. I want to help you succeed! 1. Show that \(1 + 2 + 3 + 4 + \cdots + n = \frac{n(n+1)}{2}\) for integers \(n \geq 1\) using induction. 2. Show that the Towers of Hanoi game with \(n\) disks can be solved in \(2^n - 1\) moves, with \(n \in \mathbb{Z}^+\) using induction.
Transcribed Image Text:**Learning Target R4 Core**: Given a statement to be proven by (weak) induction, I can state and prove the base case, state the inductive hypothesis, and outline the proof. I can describe the subtle difference between 'weak' and 'strong' induction. **Please remember** the academic integrity policy that you swore to uphold. Proofs can be challenging. Just try your best. If you don’t get it this week, you have many more opportunities! You might not be able to prove these claims with what we learned in this class, and that’s okay. You’re just setting it up! For each statement below, complete each of the following parts: - State and prove the base case - State the inductive hypothesis - Outline how the rest of the proof would go, that is, tell me roughly what will happen to complete the proof including the conclusion you will arrive at; you do not need to actually do the complete proof. Remember if you have a question about this, you can email/send me a message on Teams. I want to help you succeed! 1. Show that \(1 + 2 + 3 + 4 + \cdots + n = \frac{n(n+1)}{2}\) for integers \(n \geq 1\) using induction. 2. Show that the Towers of Hanoi game with \(n\) disks can be solved in \(2^n - 1\) moves, with \(n \in \mathbb{Z}^+\) using induction.
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