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27 May, 01:16

Prove that for all n ≥ 4 the inequality 2n < n! holds.

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  1. 27 May, 04:07
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    For all n ≥ 4, 2n < n!

    Step-by-step explanation:

    Let's use the induction method to prove this statement.

    In the induction method, first we prove the statement for n=4

    1) If n = 4 ⇒2 (4) < 4! ⇒2 (4) < 24 ⇒8 < 24.

    Therefore the statement holds for n=4

    2) Now we assume that the statement is valid for n = k

    ⇒2k < k!

    3) Now we will prove the statement holds for n = k + 1

    We will prove that 2 (k + 1) < (k + 1) !

    (k + 1) ! = (k+1) (k) (k-1) ... (3) (2) (1)

    If the statement is valid for k + 1, then it would mean that

    2 (k + 1) < (k+1) (k) (k-1) ... (3) (2) (1)

    2 < (k) (k-1) ... (3) (2) (1)

    which is clearly true since k ≥4

    Therefore the statement n ≥4, 2n < n! is true.
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