Explanation Given: Statements says the a + a r + a r 2 + ⋯ + a r n − 1 = a 1 − r n 1 − r if r ≠ 1 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 r + a r 2 + ⋯ + a r n − 1 = a 1 − r n 1 − r if r ≠ 1 -----(1) Step 1: Show that statement (1) is true for the natural number n = 1 . That is a = a 1 − r 1 1 − r 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 a + a r + a r 2 + ⋯ + a r k − 1 = a 1 − r k 1 − r if r ≠ 1 -----(2) Step 3: Prove that the statement is true for the next natural number k + 1 ...

11th Edition

ISBN: 9780135228982

Author: Sullivan

Publisher: PEARSON

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Chapter 12.4, Problem 30AYU

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To prove: The given statement $a + a r + a r^{2} + \dots + a r^{n - 1} = a \frac{1 - r^{n}}{1 - r}$ if $r \neq 1$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.

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Chapter 12.4, Problem 29AYU

Chapter 12.4, Problem 31AYU

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To prove: The given statement a + a r + a r 2 + ⋯ + a r n − 1 = a 1 − r n 1 − r if r ≠ 1 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 the natural number n.

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To prove: The given statement $a + a r + a r^{2} + \dots + a r^{n - 1} = a \frac{1 - r^{n}}{1 - r}$ if $r \neq 1$ 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 r + a r 2 + ⋯ + a r n − 1 = a 1 − r n 1 − r if r ≠ 1 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 the natural number n.

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To prove: The given statement a + ar + ar2 + ⋯ + arn − 1 = a1 − rn1 − r if r ≠ 1 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 $a + a r + a r^{2} + \dots + a r^{n - 1} = a \frac{1 - r^{n}}{1 - r}$ if $r \neq 1$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.