Explanation Given: 1 3 + 2 3 + 3 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n 3 = n 2 ( n + 1 ) 2 4 where n is a positive integer. 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 . It means P ( 1 ) is true. In the second step is we have to prove 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. Formula used: 1 3 + 2 3 + 3 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n 3 = n 2 ( n + 1 ) 2 4 . Calculation: Description 1 3 + 2 3 + 3 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n 3 = n 2 ( n + 1 ) 2 4 Step 1: To prove P ( 1 ) is true. P ( n ) : 1 3 + 2 3 + 3 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n 3 = n 2 ( n + 1 ) 2 4 P ( 1 ) : 1 3 = 1 2 ( 1 + 1 ) 2 4 ⇒ 1 = 2 2 4 ⇒ 1 = 4 4 ⇒ 1 = 1 Step 2: Consider P ( k ) is true. P ( k ) : 1 3 + 2 3 + 3 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + k 3 = k 2 ( k + 1 ) 2 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

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

Videos

Question

Chapter 8.5, Problem 7E

To determine

Proof for the given statement $1^{3} + 2^{3} + 3^{3} + \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot + n^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$ using mathematical induction by assuming that $n$ is a positive integer.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.5, Problem 6E

Chapter 8.5, Problem 8E

Students have asked these similar questions

Consider the table of values below. x y 2 64 3 48 4 36 5 27 Fill in the right side of the equation y= with an expression that makes each ordered pari (x,y) in the table a solution to the equation.

solving for x

Consider the table of values below. x y 2 63 3 70 4 77 5 84 Fill in the right side of the equation y= with an expression that makes each ordered pari (x,y) in the table a solution to the equation.

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.

Proof for the given statement 1 3 + 2 3 + 3 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n 3 = n 2 ( n + 1 ) 2 4 using mathematical induction by assuming that n is a positive integer.

Concept explainers

Videos

Proof for the given statement 13+23+33+⋅⋅⋅⋅⋅⋅⋅⋅+n3=n2(n+1)24 using mathematical induction by assuming that n is a positive integer.

Want to see the full answer?

Chapter 8 Solutions

Proof for the given statement $1^{3} + 2^{3} + 3^{3} + \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot + n^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$ using mathematical induction by assuming that $n$ is a positive integer.