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17 September, 01:21

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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  1. 17 September, 01:49
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    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.
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