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

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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.

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Chapter 8.5, Problem 7E

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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.

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Chapter 8.5, Problem 6E

Chapter 8.5, Problem 8E

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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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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.

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

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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.