Question:
Given an example of a statement P (n) such that it is true for all n ∈ N.
Solution:
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 $n \in \mathrm{N})$
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