Question

Let $P\left(n\right):n^2+n$ is an odd integer
$P\left(k\right)\Rightarrow P(k+1)$ is true
Then $P\left(n\right)$ is true for all

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

Answer 1

Answer: (D)

none of these

Given that $P\left(n\right):n^2+n$ is an odd integer.

$P\left(k\right)\Rightarrow P(k+1)$ is true.

For $P\left(n\right)$ to be true for all $n$, it should be true for $n=1$

$P\left(1\right)=1^2+1=2$ This is not an odd integer, So not true for $n=1$

Check for $n=2$, $P\left(2\right)=2^2+2=6$ This is again not odd,

so not true for $n=2$

For $n=3$, $P\left(3\right)=3^2+3=9$, odd so true

For $n=4$, $P\left(4\right)=4^2+4=20$ not odd, so not true.

So we see from options, it not true for any given option.