Question

$2^{n} < n!$ is true for $\left\{\text{where} \, n \in N\right\}$

• A. $n < 4$
• B. $n\geq 4$
• C. $n < 3$
• D. None of these

Answer 1

Answer: (B)

$P\left(n\right):2^{n} < n \, !$
$P\left(1\right),P\left(2\right),P\left(3\right)$ are not true.
$P\left(4\right):16 < 24$ is true
$P\left(K\right):K!>2^{K}$ is true for some $K\in N, \, K>4$
To prove $P\left(K + 1\right):\left(K + 1\right)!>2^{K + 1}$ is true
Multiplying both sides of $P\left(K\right)$ by $\left(K + 1\right),$ we get,
$\left(K + 1\right)!>2^{K}\left(K + 1\right)>2^{K}.2\left(as K > 4\right)$
Hence by Principle of Mathematical induction, $P\left(n\right)$ is true $\forall \, n\in N, \, \, n\geq 4$