Question

$\text{If}P\left(n\right):2^n<n!$, then the smallest positive integer for which $P\left(n\right)$ is true, is

• A. $4$
• B. $2$
• C. $5$
• D. $3$

Answer 1

The correct option is A $4$

$\text{For}n=1$,
$P\left(1\right):2^1\nless 1$
$\text{For}n=2$,
$P\left(2\right):2^2\nless 2$
$\text{For}n=3$,
$P\left(3\right):2^3\nless 3!=6$
$\text{For}n=4$,
$P\left(4\right):2^4<4!=24$
Therefore, the smallest positive integer for which $P\left(n\right)$ is true is $n=4$.