Question 31 pts Prove the statement is true using mathematical induction: 2n-1 ≤ n! Use the space below to write your answer. To make the < symbol, you might want to use the < with the underline feature.

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Mathematics

Katrina Carr

2 August, 13:28

Question 31 pts Prove the statement is true using mathematical induction: 2n-1 ≤ n! Use the space below to write your answer. To make the < symbol, you might want to use the < with the underline feature.

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Answers (1)

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Elliana Wang · Answer 1

P (3) is true since 2 (3) - 1 = 5 < 3! = 6.

Step-by-step explanation:

Let P (n) be the proposition that 2n-1 ≤ n!. for n ≥ 3

Basis: P (3) is true since 2 (3) - 1 = 5 < 3! = 6.

Inductive Step: Assume P (k) holds, i. e., 2k - 1 ≤ k! for an arbitrary integer k ≥ 3. To show that P (k + 1) holds:

2 (k+1) - 1 = 2k + 2 - 1

≤ 2 + k! (by the inductive hypothesis)

= (k + 1) ! Therefore, 2n-1 ≤ n! holds, for every integer n ≥ 3.