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The inductive step of an inductive proof shows that for k ≥ 4, if 2k > 3k, then 2k+1 ≥ 3(k+1). In which step uses the fact that k> 4> 1? a. Step 2 b. Step 3 c. Step 4 d. Step 5 2k+1 > 2.2k 2k+1 > 2.3k 2k+13k+ 3k 2k+1 ≥ 3k +3 2k+1 ≥3(k+1) (Step 1) (Step 2) (Step 3) (Step 4) (Step 5)

The inductive step of an inductive proof shows that for k ≥ 4, if 2k > 3k, then 2k+1 ≥ 3(k+1). In which step uses the fact that k> 4> 1? a. Step 2 b. Step 3 c. Step 4 d. Step 5 2k+1 > 2.2k 2k+1 > 2.3k 2k+13k+ 3k 2k+1 ≥ 3k +3 2k+1 ≥3(k+1) (Step 1) (Step 2) (Step 3) (Step 4) (Step 5)

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:** The inductive step of an inductive proof shows that for \( k \geq 4 \), if \( 2^k \geq 3k \), then \( 2^{k+1} \geq 3(k + 1) \). In which step uses the fact that \( k \geq 4 \geq 1 \)? **Steps:** 1. \( 2^{k+1} \geq 2 \cdot 2^k \) \hspace{20pt} (Step 1) 2. \( 2^{k+1} \geq 2 \cdot 3k \) \hspace{20pt} (Step 2) 3. \( 2^{k+1} \geq 3k + 3k \) \hspace{20pt} (Step 3) 4. \( 2^{k+1} \geq 3k + 3 \) \hspace{20pt} (Step 4) 5. \( 2^{k+1} \geq 3(k + 1) \) \hspace{20pt} (Step 5) **Options:** - a. Step 2 - b. Step 3 - c. Step 4 - d. Step 5 **Explanation:** The inductive hypothesis is applied in one of these steps to use the fact that \( k \geq 4 \) which allows simplifying or validating the inequality as part of the deduction process. Each step logically follows from the previous assumptions and algebraic transformations. To determine which step directly involves the fact that \( k \geq 4 \), analyze how the inductive hypothesis is utilized.
Transcribed Image Text:**Problem Statement:** The inductive step of an inductive proof shows that for \( k \geq 4 \), if \( 2^k \geq 3k \), then \( 2^{k+1} \geq 3(k + 1) \). In which step uses the fact that \( k \geq 4 \geq 1 \)? **Steps:** 1. \( 2^{k+1} \geq 2 \cdot 2^k \) \hspace{20pt} (Step 1) 2. \( 2^{k+1} \geq 2 \cdot 3k \) \hspace{20pt} (Step 2) 3. \( 2^{k+1} \geq 3k + 3k \) \hspace{20pt} (Step 3) 4. \( 2^{k+1} \geq 3k + 3 \) \hspace{20pt} (Step 4) 5. \( 2^{k+1} \geq 3(k + 1) \) \hspace{20pt} (Step 5) **Options:** - a. Step 2 - b. Step 3 - c. Step 4 - d. Step 5 **Explanation:** The inductive hypothesis is applied in one of these steps to use the fact that \( k \geq 4 \) which allows simplifying or validating the inequality as part of the deduction process. Each step logically follows from the previous assumptions and algebraic transformations. To determine which step directly involves the fact that \( k \geq 4 \), analyze how the inductive hypothesis is utilized.
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