For the following statements: • State the base case and prove that it is true. • State the inductive hypothesis • Outline how would you proceed with the rest of the proof. Explain roughly what will exactly happen to complete the proof. It's not actually required to complete the proof

For the following statements: • State the base case and prove that it is true. • State the inductive hypothesis • Outline how would you proceed with the rest of the proof. Explain roughly what will exactly happen to complete the proof. It's not actually required to complete the proof

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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Question

For the following statements:

• State the base case and prove that it is true.

• State the inductive hypothesis

• Outline how would you proceed with the rest of the proof. Explain roughly what will exactly happen to complete the proof. It's not actually required to complete the proof.

You are *setting up* the inductive proofs.

1. Prove that \(1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\)

2. Prove that if \(x \in \mathbb{R}\) and \(x > -1\), then \((1 + x)^n \geq 1 + nx\) for all \(n \in \mathbb{N}\)

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