Explanation Given: 3 n > 2 n + 1 for n ≥ 2 First we have to consider the given statement as P ( n ) . P ( n ) , Proof of Induction is a mathematical statement involving a value n . By Proof of Induction, in the first step we have to prove P ( n ) is true for n = 1 . But here in this problem it is given that n ≥ 2 . Therefore here we have to prove P ( n ) is true for n = 2 . It means P ( 2 ) is true. In the second step is we have to consider P ( n ) is true for n = k , where k is any value. It means P ( k ) is true. This is called Induction Hypothesis. We will now prove directly that P ( k ) ⇒ P ( k + 1 ) , which means P ( k + 1 ) is true by using the fact that P ( k ) is true. This is called Induction Step. The combination of these steps shows that P ( n ) is true for all values of n. For Left Hand Side Expression> Right Hand Side Expression inequality statements, in the third step we have to prove that Difference of Right Hand Side expression and Left Hand Side is negative. For Left Hand Side Expression < Right Hand Side Expression inequality statements, in the third step we have to prove that Difference of Right Hand Side expression and Left Hand Side is positive. Formula used: 3 n > 2 n + 1 for n ≥ 2 Calculation: Description 3 n > 2 n + 1 for n ≥ 2 Step 1: To prove P ( 2 ) is true. P ( n ) : 3 n > 2 n + 1 P ( 2 ) : 3 2 > 2 ( 3 ) + 1 ⇒ 9 > 7 Step 2: Consider P ( k ) is true...

College Algebra with Modeling & Visualization (5th Edition)

5th Edition

ISBN: 9780321826138

Author: Gary K. Rockswold

Publisher: PEARSON

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Chapter 8.5, Problem 22E

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The proof that $3^{n} > 2 n + 1$ for $n \geq 2$ by using mathematical induction.

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Chapter 8.5, Problem 21E

Chapter 8.5, Problem 23E

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Chapter 8 Solutions

College Algebra with Modeling & Visualization (5th Edition)

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

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The proof that 3 n &gt; 2 n + 1 for n ≥ 2 by using mathematical induction.

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The proof that 3 n > 2 n + 1 for n ≥ 2 by using mathematical induction.

The proof that $3^{n} > 2 n + 1$ for $n \geq 2$ by using mathematical induction.