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Prove that 3" +4" ≤5" for all integers n ≥ 2.

Prove that 3" +4" ≤5" for all integers n ≥ 2.

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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**Problem Statement:** Prove that \(3^n + 4^n \leq 5^n\) for all integers \(n \geq 2\). **Detailed Explanation:** This inequality involves exponential functions with bases 3, 4, and 5. The goal is to demonstrate that the sum of the exponential terms with bases 3 and 4 does not exceed the exponential term with base 5 for all integer values of \(n\) equal to or greater than 2. A common method for proving such inequalities is mathematical induction. This involves two main steps: 1. **Base Case:** Verify that the inequality holds for the initial value of \(n\), which in this case is \(n = 2\). 2. **Inductive Step:** Assume that the inequality holds for some arbitrary integer \(k \geq 2\), i.e., \(3^k + 4^k \leq 5^k\). Then show that this assumption implies the inequality holds for \(k + 1\), i.e., \(3^{k+1} + 4^{k+1} \leq 5^{k+1}\). As you work through the problem, consider the behavior of exponential growth and how the choice of bases affects the inequality. The exponential term \(5^n\) grows faster than \(3^n\) and \(4^n\), providing the intuition behind the inequality's truth for sufficiently large values of \(n\).
Transcribed Image Text:**Problem Statement:** Prove that \(3^n + 4^n \leq 5^n\) for all integers \(n \geq 2\). **Detailed Explanation:** This inequality involves exponential functions with bases 3, 4, and 5. The goal is to demonstrate that the sum of the exponential terms with bases 3 and 4 does not exceed the exponential term with base 5 for all integer values of \(n\) equal to or greater than 2. A common method for proving such inequalities is mathematical induction. This involves two main steps: 1. **Base Case:** Verify that the inequality holds for the initial value of \(n\), which in this case is \(n = 2\). 2. **Inductive Step:** Assume that the inequality holds for some arbitrary integer \(k \geq 2\), i.e., \(3^k + 4^k \leq 5^k\). Then show that this assumption implies the inequality holds for \(k + 1\), i.e., \(3^{k+1} + 4^{k+1} \leq 5^{k+1}\). As you work through the problem, consider the behavior of exponential growth and how the choice of bases affects the inequality. The exponential term \(5^n\) grows faster than \(3^n\) and \(4^n\), providing the intuition behind the inequality's truth for sufficiently large values of \(n\).
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