Explanation Given: Statements says the a + ( a + d ) + ( a + 2 d ) + ⋯ + [ a + ( n − 1 ) d ] = n a + d n ( n − 1 ) 2 is true for all natural number. Formula used: The Principle of Mathematical Induction Suppose that the following two conditions are satisfied with regard to a statement about natural numbers: CONDITION I: The statement is true for the natural number 1. CONDITION II: If the statement is true for some natural number k , it is also true for the next natural number k + 1 . Then the statement is true for all natural numbers. Proof: Consider the statement a + ( a + d ) + ( a + 2 d ) + ⋯ + [ a + ( n − 1 ) d ] = n a + d n ( n − 1 ) 2 -----(1) Step 1: Show that statement (1) is true for the natural number n = 1 . That is a + ( 1 − 1 ) d = 1. a + d 1 ( 1 − 1 ) 2 = a . Hence the statement is true for the natural number n = 1 . Step 2: Assume that the statement is true for some natural number k . That is a + ( a + d ) + ( a + 2 d ) + ⋯ + [ a + ( k − 1 ) d ] = k a + d k ( k − 1 ) 2 -----(2) Step 3: Prove that the statement is true for the next natural number k + 1 ...

11th Edition

ISBN: 9780135189405

Author: Michael Sullivan

Publisher: PEARSON

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Chapter 12.4, Problem 31AYU

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To prove: The given statement $a + (a + d) + (a + 2 d) + \dots + [a + (n - 1) d] = n a + d \frac{n (n - 1)}{2}$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.

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Chapter 12.4, Problem 30AYU

Chapter 12.4, Problem 32AYU

Chapter 12 Solutions

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To prove: The given statement a + ( a + d ) + ( a + 2 d ) + ⋯ + [ a + ( n − 1 ) d ] = n a + d n ( n − 1 ) 2 is true for all natural numbers n using the Principle of Mathematical Induction.

Solution Summary: The author explains the Principle of Mathematical Induction. The statement is true for all natural numbers.

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To prove: The given statement $a + (a + d) + (a + 2 d) + \dots + [a + (n - 1) d] = n a + d \frac{n (n - 1)}{2}$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.

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Chapter 12 Solutions

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To prove: The given statement a + ( a + d ) + ( a + 2 d ) + ⋯ + [ a + ( n − 1 ) d ] = n a + d n ( n − 1 ) 2 is true for all natural numbers n using the Principle of Mathematical Induction.

Solution Summary: The author explains the Principle of Mathematical Induction. The statement is true for all natural numbers.

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To prove: The given statement a + (a + d) + (a + 2d) + ⋯ + [ a + (n − 1)d ] = na + dn(n − 1)2 is true for all natural numbers n using the Principle of Mathematical Induction.

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Chapter 12 Solutions

Additional Math Textbook Solutions

To prove: The given statement $a + (a + d) + (a + 2 d) + \dots + [a + (n - 1) d] = n a + d \frac{n (n - 1)}{2}$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.