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Prove that for all natural numbers n ≥ 5, (n + 1)! > 2n+3.

Prove that for all natural numbers n ≥ 5, (n + 1)! > 2n+3.

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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**Mathematical Induction Problem:** Prove that for all natural numbers \( n \geq 5 \), the inequality \((n + 1)! > 2^{n+3}\) holds true. **Explanation:** This is a mathematical problem involving factorials and powers of two. The problem statement asks us to demonstrate that for any natural number \( n \) starting from 5, the factorial of \( n+1 \) is greater than \( 2 \) raised to the power of \( n+3 \). A common method to prove such statements is by using mathematical induction. This involves two main steps: 1. **Base Case:** Verify the statement is true for the starting point (e.g., \( n = 5 \)). 2. **Inductive Step:** Assume it is true for some \( n = k \) and then prove it is true for \( n = k + 1 \). This method helps establish the truth of the inequality for all \( n \geq 5 \).
Transcribed Image Text:**Mathematical Induction Problem:** Prove that for all natural numbers \( n \geq 5 \), the inequality \((n + 1)! > 2^{n+3}\) holds true. **Explanation:** This is a mathematical problem involving factorials and powers of two. The problem statement asks us to demonstrate that for any natural number \( n \) starting from 5, the factorial of \( n+1 \) is greater than \( 2 \) raised to the power of \( n+3 \). A common method to prove such statements is by using mathematical induction. This involves two main steps: 1. **Base Case:** Verify the statement is true for the starting point (e.g., \( n = 5 \)). 2. **Inductive Step:** Assume it is true for some \( n = k \) and then prove it is true for \( n = k + 1 \). This method helps establish the truth of the inequality for all \( n \geq 5 \).
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