Principle Of Mathematical Induction Question 50

Question: If $ P(n):3^{n}<n!,n\in N, $ then P(n) is true

Options:

A) For $ n\ge 6 $

B) For $ n\ge 7 $

C) For $ n\ge 3 $

D) For all n.

Show Answer

Answer:

Correct Answer: B

Solution:

  • [b] Given that, $ P(n):3^{n}<n! $ Now, $ P(7):3^{7}<7! $ is true Let $ P(k):3^{k}<k! $
    $ \Rightarrow P(k+1):{3^{k+1}} $ $ ={{3.3}^{k}}<3.k!<(k+1)!(\because k+1>3) $