Question

Let $P\left(n\right):n^2+n$ is an odd integer. It is seen that truth of $P\left(n\right)\Rightarrow$ the truth of P(n + 1). Therefore, P(n) is true for all

• A. n > 1
• B. n
• C. n > 2
• D. none of these

Answer 1

The correct option is D none of these

$P\left(n\right)=n^2+n$
$=n(n+1)$
Hence $P\left(n\right)$ represents product of two consecutive Natural numbers.
We know that every odd natural number is succeeded by an even natural number. Hence product of two natural numbers will always be even.
Hence
$P\left(n\right)$ is even for all $n\in N$.
Hence the conclusion that $P\left(n\right)=n^2+n$ is an odd integer is false.