Give an example of a statement P(n)

According to the question,

P(n) which is true for all n.

Let P(n) be,

$1+2+3+\cdots \ldots \ldots+n=\frac{n(n+1)}{2}$

$\mathrm{P}(0)$ is $0=\frac{0(0+1)}{2}=0 ;$ it's true

$\mathrm{P}(1)$ is $1=\frac{1(1+1)}{2}=1 ;$ it's true

$P(2)$ is $1+2=\frac{2(2+1)}{2} ;$ it's true

$P(k)$ is $1+2+3+\cdots \ldots \ldots+k=\frac{k(k+1)}{2}$

$\mathrm{P}(\mathrm{k})$ is $1+2+3+\cdots \ldots \ldots+\mathrm{k}+1=\frac{\mathrm{k}(\mathrm{k}+1)}{2}+\mathrm{k}+1=\frac{(\mathrm{k}+1)(\mathrm{k}+2)}{2}$

⇒ P(k) is true for all k.

Therefore, P(n) is true for all n.