Use mathematical induction to show that 32n- 1 is divisible by 8 for all natural numbers n.Let P(n) denote the statement that 32n – 1 is divisible by 8.-P(1) is the statement that 3- 1 =is divisible by 8, which is true.Assume that P(k) is true. Thus, our induction hypothesis isis divisible by 8.We want to use this to show that P(k + 1) is true. Now,32(k +9 3- 1- 19 3+ 89 32k+ 8.This final result is divisible by 8, sinceis divisible by 8 by the induction hypothesis. Thus, P(k + 1) follows from P(k), and this completes the induction step.Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is true for all natural numbers n.

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Description: Use mathematical induction to show that 32n- 1 is divisible by 8 for all natural numbers n.Let P(n) denote the statement that 32n – 1 is divisible by 8.-P(1) is the statement that 3- 1 =is divisible by 8, which is true.Assume that P(k) is true. Thus

Let P(n): 32n – 1 is divisible by 8. P(1): 32 – 1 = 8 which is divisible by 8. Thus P(n) is true…

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Use mathematical induction to show that 34n - 1 is divisible by 8 for all natural numbers n. Let P(n) denote the statement that 32n - 1 is divisible by 8. P(1) is the statement that is divisible by 8, which is true. Assume that P(k) is true. Thus, our induction hypothesis is is divisible by8. We want to use this to show that P(k + 1) is true. Now, 32(k + - 1 = 9 + 8 - = 9 34K - + 8. This final result is divisible by 8, since is divisible by 8 by the induction hypothesis. Thus, P(k + 1) follows from P(k), and this completes the induction step. Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is true for all natural numbers n.

Use mathematical induction to show that 34n - 1 is divisible by 8 for all natural numbers n. Let P(n) denote the statement that 32n - 1 is divisible by 8. P(1) is the statement that is divisible by 8, which is true. Assume that P(k) is true. Thus, our induction hypothesis is is divisible by8. We want to use this to show that P(k + 1) is true. Now, 32(k + - 1 = 9 + 8 - = 9 34K - + 8. This final result is divisible by 8, since is divisible by 8 by the induction hypothesis. Thus, P(k + 1) follows from P(k), and this completes the induction step. Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is true for all natural numbers n.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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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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Use mathematical induction to show that 32n
- 1 is divisible by 8 for all natural numbers n.
Let P(n) denote the statement that 32n – 1 is divisible by 8.
-
P(1) is the statement that 3
- 1 =
is divisible by 8, which is true.
Assume that P(k) is true. Thus, our induction hypothesis is
is divisible by 8.
We want to use this to show that P(k + 1) is true. Now,
32(k +
9 3
- 1
- 1
9 3
+ 8
9 32k
+ 8.
This final result is divisible by 8, since
is divisible by 8 by the induction hypothesis. Thus, P(k + 1) follows from P(k), and this completes the induction step.
Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is true for all natural numbers n.

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