Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.* 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, identifying how this section of the proof will begin and ultimately what it will look like when you’re done (as modeled in class videos); you do not need to actually do the complete proof. 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! 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 \( 2! \cdot 4! \cdot 6! \cdot \ldots \cdot (2n)! \geq ((n+1)!)^n \) for \( n \in \mathbb{Z}^+ \). 2. Show that \( a + (a+d) + (a+2d) + (a+3d) + \ldots + (a+(n-1)d) = \frac{n(2a+(n-1)d)}{2} \) for \( n \in \mathbb{Z}^+ \).
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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.* 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, identifying how this section of the proof will begin and ultimately what it will look like when you’re done (as modeled in class videos); you do not need to actually do the complete proof. 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! 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 \( 2! \cdot 4! \cdot 6! \cdot \ldots \cdot (2n)! \geq ((n+1)!)^n \) for \( n \in \mathbb{Z}^+ \). 2. Show that \( a + (a+d) + (a+2d) + (a+3d) + \ldots + (a+(n-1)d) = \frac{n(2a+(n-1)d)}{2} \) for \( n \in \mathbb{Z}^+ \).
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