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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. 2 P(1) is the statement that 3 - 18 is divisible by 8, which is true. Assume that P(k) is true. Thus, our induction hypothesis is We want to use this to show that P(k + 1) is true. Now, 32(k+ X ) = 93 × k "). 1 = 93 x k X +8 9(32K - × ) + 8. is divisible by 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 32n - 1 is divisible by 8 for all natural numbers n. Let P(n) denote the statement that 32n 1 is divisible by 8. 2 P(1) is the statement that 3 - 18 is divisible by 8, which is true. Assume that P(k) is true. Thus, our induction hypothesis is We want to use this to show that P(k + 1) is true. Now, 32(k+ X ) = 93 × k "). 1 = 93 x k X +8 9(32K - × ) + 8. is divisible by 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.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 42E
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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. 2 P(1) is the statement that 3 - 18 is divisible by 8, which is true. Assume that P(k) is true. Thus, our induction hypothesis is We want to use this to show that P(k + 1) is true. Now, 32(k+ X ) = 93 × k "). 1 = 93 x k X +8 9(32K - × ) + 8. is divisible by 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.
Transcribed Image Text: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. 2 P(1) is the statement that 3 - 18 is divisible by 8, which is true. Assume that P(k) is true. Thus, our induction hypothesis is We want to use this to show that P(k + 1) is true. Now, 32(k+ X ) = 93 × k "). 1 = 93 x k X +8 9(32K - × ) + 8. is divisible by 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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