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Question
Given an example of a statement P (n) such that it is true for all n ∈ N.
Solution
Proved: \[P(n) = n^2 + n \text{ is even for } P(n) \text{ and } P(n + 1) . \]
\[\text{ Therefore } , \frac{n^2 + n}{2} \text{ is also even for all n} \in N . \left[ \text{ Dividing an even number by 2 gives an even number } . \right]\]
\[\text{ Thus, we have: } \]
\[P(n) = 1 + 2 + . . . + n \]
\[ = \frac{n(n + 1)}{2} (\text{ Even for all n } \in N)\]
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