Explanation Given: Statements says the series ( 1 + a ) n ≥ 1 + n a , for a > 0 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 ( 1 + a ) n ≥ 1 + n a for a > 0 -----(1) Step 1: Show that statement (1) is true for the natural number n = 1 . That is ( 1 + a ) 1 = 1 + a = 1 + 1. 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 ( 1 + a ) k ≥ 1 + k a for a > 0 -----(2) Step 3: Prove that the statement is true for the next natural number k + 1 ...

10th Edition

ISBN: 9780321979070

Author: Michael Sullivan

Publisher: PEARSON

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Chapter 12.4, Problem 27AYU

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To prove: The given statement ${(1 + a)}^{n} \geq 1 + n a,$ for $a > 0$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.

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Chapter 12.4, Problem 26AYU

Chapter 12.4, Problem 28AYU

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To prove: The given statement ( 1 + a ) n ≥ 1 + n a , for a > 0 is true for all natural numbers n using the Principle of Mathematical Induction.

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

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To prove: The given statement ${(1 + a)}^{n} \geq 1 + n a,$ for $a > 0$ 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 ( 1 + a ) n ≥ 1 + n a , for a &gt; 0 is true for all natural numbers n using the Principle of Mathematical Induction.

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

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To prove: The given statement (1 + a)n ≥ 1 + na, for a > 0 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 ( 1 + a ) n ≥ 1 + n a , for a > 0 is true for all natural numbers n using the Principle of Mathematical Induction.

To prove: The given statement ${(1 + a)}^{n} \geq 1 + n a,$ for $a > 0$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.