**Problem 9: Proof by Induction****Statement:** Prove by induction: If \( n \in \mathbb{N} \), then \( n < 2^n \).**Explanation:**The statement is asking you to demonstrate, using mathematical induction, that for every natural number \( n \), the inequality \( n < 2^n \) holds true. **Key Concepts:**- **Mathematical Induction:** A proof technique used to demonstrate the truth of an infinite number of cases. It consists of two main steps: 1. *Base Case:* Prove the statement for the initial value (usually \( n = 1 \)). 2. *Inductive Step:* Assume the statement is true for \( n = k \), and then prove it is true for \( n = k + 1 \).**Procedure:**1. **Base Case**: Verify the statement for the smallest natural number. - Show that \( 1 < 2^1 \).2. **Inductive Step**: Assume the statement is true for \( n = k \), meaning \( k < 2^k \). - Now show that \( k + 1 < 2^{k+1} \).This approach systematically demonstrates the inequality for all natural numbers.

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Math

Advanced Math

Prove by induction: If n € N, then n < 2"

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

**Problem 9: Proof by Induction**

**Statement:** Prove by induction: If \( n \in \mathbb{N} \), then \( n < 2^n \).

**Explanation:**

The statement is asking you to demonstrate, using mathematical induction, that for every natural number \( n \), the inequality \( n < 2^n \) holds true.

**Key Concepts:**

- **Mathematical Induction:** A proof technique used to demonstrate the truth of an infinite number of cases. It consists of two main steps:
1. *Base Case:* Prove the statement for the initial value (usually \( n = 1 \)).
2. *Inductive Step:* Assume the statement is true for \( n = k \), and then prove it is true for \( n = k + 1 \).

**Procedure:**

1. **Base Case**: Verify the statement for the smallest natural number.
- Show that \( 1 < 2^1 \).

2. **Inductive Step**: Assume the statement is true for \( n = k \), meaning \( k < 2^k \).
- Now show that \( k + 1 < 2^{k+1} \).

This approach systematically demonstrates the inequality for all natural numbers.

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