Question

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

Answer 1

Let the statement P(n) : 3n < n!

For $n=1,3\times 1<1!$

For $n=2,3\times 2<2!\Rightarrow 6<2\left[false\right]$

For $n=3,3\times 3<3!\Rightarrow 9<6\left[false\right]$

For $n=4,3\times 4<4!\Rightarrow 12<24\left[false\right]$

For $n=5,3\times 5<5!\Rightarrow 15<5\times 4\times 3\times 2\times 1$

$\Rightarrow 15<120\left[True\right]$