Question

Give an example of a statement P(n) which is true for all $n\ge 4$ but P(1), P(2) and P(3) are not true. Justify your answer.

Answer 1

Let $P\left(n\right):2^n<n!$

$n=1\Rightarrow 2^1<1!\Rightarrow 2<1$ [False]

$n=2\Rightarrow 2^2<2!\Rightarrow 4<4$ [False]

$n=3\Rightarrow 2^3<3!\Rightarrow 8<6$ [False]

$n=4\Rightarrow 2^4<4!\Rightarrow 16<24$ [True]

$n=5\Rightarrow 2^5<5!\Rightarrow 32<120$[True]

$n=6\Rightarrow 2^6<6!\Rightarrow 64<720$ [True]

Now, as we are multiplying LHS by 2 and RHS
by n, so LHS will always be less than RHS.

$\Rightarrow 2^n<n!$

Clearly here P(n) is true for all $n\ge 4$

Hence, the example of P(n) is$2^n<n!$