Question

lf $P\left(n\right)$ be the statement $n(n+1)+1$, then which of the following is even ?

• A. $P\left(2\right)$
• B. $P\left(3\right)$
• C. $P\left(4\right)$
• D. None of the above is even

Answer 1

The correct option is D None of the above is even

Case I
If $n$ is odd, then $n+1$ will be even. This implies,
$\left(n\right)(n+1)$ will be even. ...(since (n+1) is even).
Therefore the expression $\left(n\right)(n+1)+1$ will be odd ...(even no. +1).
Case II
If n is even, then n+1 will be odd. This implies,
$\left(n\right)(n+1)$ will be even. ...(since (n) is even).
Therefore the expression $\left(n\right)(n+1)+1$ will be odd ...(even no. +1).
Hence for all $nϵN$ the $n(n+1)+1$ is odd.