Explanation Given: The statement 3 i s a f a c t o r o f n ( n + 1 ) ( n − 1 ) Formula used: Mathematical induction is process of proving a theorem by showing that if it is true of any particular case, then it is true of the next case in a series and then showing that it is actually true in one particular case. The principle of mathematical induction provides a method to prove statements about the natural numbers are true for all natural numbers. First step in a mathematical induction proof that a statement P n is true for all natural numbers n is that P 1 is true. Second step in a mathematical induction proof is to assume that P k is true for all natural numbers k and then show that P k + 1 is true. Calculation: Given the statement 3 i s a f a c t o r o f n ( n + 1 ) ( n − 1 ) First step to show that the above statement is true for n = 1 . Substituting n with 1 , we have We have 1 ( 1 + 1 ) ( 1 − 1 ) = 1 ( 2 ) ( 0 ) = 0 3 is factor 0 The given statement is true for n = 1 . Now let us assume that the statement is true for natural number n = k . Thus, replacing n with k , we get P k :3 i s a f a c t o r o f k ( k + 1 ) ( k − 1 ) k ( k + 1 ) ( k − 1 ) = 3 m , where m is any positive integer k 3 − k = 3 m Now on that basis of the fact that P k is true, show that P k + 1 is true

Precalculus: A Unit Circle Approach, Books a la Carte Edition plus MyMathLab with Pearson eText -- Access Card Package (2nd Edition)

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ISBN: 9780321869401

Author: J. S. Ratti

Publisher: PEARSON

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Chapter 9.4, Problem 34E

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To prove:

The statement 3 is a factor of n(n+1)(n−1) is true for all natural numbers using mathematical induction.

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Chapter 9.4, Problem 33E

Chapter 9.4, Problem 35E

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u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (ū+v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅w) Support your answer mathematically or a with a written explanation. d) If possible, find u. (vxw) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Question 3 (6 points) u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (u + v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅ w) Support your answer mathematically or a with a written explanation. d) If possible, find u (v × w) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

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Precalculus: A Unit Circle Approach, Books a la Carte Edition plus MyMathLab with Pearson eText -- Access Card Package (2nd Edition)

Ch. 9.1 - An infinite sequence is a function whose domain is...Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - In Exercises 9-28, write the first four terms of...Ch. 9.1 - In Exercises 9-28, write the first four terms of...Ch. 9.1 - In Exercises 9-28, write the first four terms of...Ch. 9.1 - In Exercises 9-28, write the first four terms of...

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To prove: The statement 3 i s a f a c t o r o f n ( n + 1 ) ( n − 1 ) is true for all natural numbers using mathematical induction.

Solution Summary: The author explains how mathematical induction proves a theorem by showing that it is true for all natural numbers.

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To prove:

The statement 3 is a factor of n(n+1)(n−1) is true for all natural numbers using mathematical induction.

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Chapter 9 Solutions

To prove: The statement 3 i s a f a c t o r o f n ( n + 1 ) ( n − 1 ) is true for all natural numbers using mathematical induction.

Solution Summary: The author explains how mathematical induction proves a theorem by showing that it is true for all natural numbers.

Concept explainers

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To prove: The statement 3 is a factor of n(n+1)(n−1) is true for all natural numbers using mathematical induction.

Want to see the full answer?

Chapter 9 Solutions

To prove:

The statement 3 is a factor of n(n+1)(n−1) is true for all natural numbers using mathematical induction.