First slide

Introduction to P.M.I

Question

Let p(n) be statement $2^{n} < n!$ , where n is a natural number, then p(n) is true for:

Moderate

A

all n
B

all n $n > 2$
C

all $n > 3$
D

None of these

Solution

Let, $p (n) : 2^{n} < n!$
then, $p (1) : 2! < 1!,$ which is not true
   $p (2) : 2^{2} < 2!$ which is not true
   $p (3) : 2^{3} < 3!$ which is not true
   $p (4) : 2^{4} < 4!$ which is true.
let $p (k) be true if k \geq 4, i.e., 2^{k} < k!, k \geq 4$
$\begin{aligned} \Rightarrow 2 {.2}^{k} < 2 (k!) \Rightarrow 2^{k + 1} < k (k!) (∵ k \geq 4 > 2) \\ \Rightarrow 2^{k + 1} (k + 1)! \Rightarrow p (k + 1) is true. \end{aligned}$
Hence, we conclude that p(n) is not true for n = 2,3 but
hold true for n ≥ 4.

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