Explanation Given: 4 n > n 4 for n ≥ 5 . 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 = 5 . But here in this problem it is given that n > 4 . Therefore here we have to prove P ( n ) is true for n = 5 . It means P ( 5 ) 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: 4 n > n 4 for n ≥ 5 . Calculation: Description 4 n > n 4 for n ≥ 5 Step 1: To prove P ( 5 ) is true. P ( n ) : 4 n > n 4 P ( 5 ) : 4 5 > 5 4 ⇒ 1024 > 625 Step 2: Consider P ( k ) is true. P ( k ) : 4 k > k 4 Step3: To prove P ( k + 1 ) is true by considering P ( k ) is true...

College Algebra with Modeling & Visualization (5th Edition)

5th Edition

ISBN: 9780321826138

Author: Gary K. Rockswold

Publisher: PEARSON

See similar textbooks

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Videos

Question

Chapter 8.5, Problem 28E

To determine

The proof that $4^{n} > n^{4}$ for $n \geq 5$ by using mathematical induction.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.5, Problem 27E

Chapter 8.5, Problem 29E

Students have asked these similar questions

How long is a guy wire reaching from the top of a 15-foot pole to a point on the ground 9-feet from the pole? Question content area bottom Part 1 The guy wire is exactly feet long. (Type an exact answer, using radicals as needed.) Part 2 The guy wire is approximatelyfeet long. (Round to the nearest thousandth.)

Question 6 Not yet answered Marked out of 5.00 Flag question = If (4,6,-11) and (-12,-16,4), = Compute the cross product vx w k

Consider the following vector field v^-> (x,y): v^->(x,y)=2yi−xj What is the magnitude of the vector v⃗ located in point (13,9)? [Provide your answer as an integer number (no fraction). For a decimal number, round your answer to 2 decimal places]

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

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The proof that 4 n &gt; n 4 for n ≥ 5 by using mathematical induction.

Concept explainers

Videos

The proof that 4n>n4 for n≥5 by using mathematical induction.

Want to see the full answer?

Chapter 8 Solutions

The proof that 4 n > n 4 for n ≥ 5 by using mathematical induction.

The proof that $4^{n} > n^{4}$ for $n \geq 5$ by using mathematical induction.