Question

lf $P\left(n\right)$ be the statement $n(n+1),n\in N$ is even, then

• A. $P\left(2\right)$ is true
• B. $P\left(3\right)$ is true
• C. $P\left(4\right)$ is true
• D. all of the above

Answer 1

The correct option is D all of the above

Case 1. If $n$ is even $\Rightarrow n+1$ will be an odd
Thus $P\left(n\right)=n(n+1)=$ even $\times$ odd $=$even
Case 2. If $n$ is odd $\Rightarrow n+1$ will be an even
Thus $P\left(n\right)=n(n+1)=$ odd $\times$ even $=$even
Hence $P\left(n\right)$ is an even number for all $n\in N$
In particular $P\left(2\right),P\left(3\right),P\left(4\right)$ are even