Question

Let $P\left(n\right)$ be the statement $2^n<n!$ where $n$ is a natural number, then $P\left(n\right)$ is true for:

• A. all $n$
• B. all $n>2$
• C. all $n>3$
• D. all $n<3$

Answer 1

The correct option is C all $n>3$

We have,

$P\left(n\right)$ be the statement $2^n<n!$

Where $n$ is a natural number

Then,

Put $n=1,2,\mathrm{3....}$

So,

$P\left(1\right)$ be the statement of $2^1<1!=2<1\left(\text{It}\text{is}\text{wrong}\right)$

$P\left(2\right)$ be the statement of $2^2<2!=4<2\left(\text{It}\text{is}\text{wrong}\right)$

$P\left(3\right)$ be the statement of $2^3<3!=8<6\left(\text{It}\text{is}\text{wrong}\right)$

But,

$P\left(4\right)$ be the statement of $2^4<4!=16<24\left(\text{It}\text{is}\text{right}\right)$

Similarly,

$P\left(5\right),P\left(6\right).......$ So, all number $n$ is a natural number.

Therefore, it is true for all $n>3$.

Hence, this is the answer.