Given an example of a statement P (n) such that it is true for all

Proved:

$P(n)=n^{2}+n$ is even for $P(n)$ and $P(n+1)$.

Therefore, $\frac{n^{2}+n}{2}$ is also even for all $n \in N$.

[Dividing an even number by 2 gives an even number.]

Thus, we have :

$P(n)=1+2+\ldots+n$

$=\frac{n(n+1)}{2} \quad($ Even for all $n \in \mathrm{N})$