Question

Let $P\left(n\right)=1+3+5+....+(2n-1)=3+n^2,n\in N$
Statement $1$ : The statement $P\left(n\right)$ is true
Statement $2$ : According to the principle of mathematical induction, if $P\left(k\right)\Rightarrow P(k+1)$ is true, then the given statement $P\left(n\right)$ is true

• A. The statement $1$ is correct and the statement $2$ is incorrect
• B. The statement $1$ is incorrect and the statement $2$ is correct
• C. Both the statements are correct
• D. Both the statements are incorrect

Answer 1

The correct option is D Both the statements are incorrect

Any statement $P$ which is defined for natural numbers (subset of natural numbers), we can check the validity of the statement using $\text{PMI}$ as following
$\left(1\right)$ The base case must be true
$\left(2\right)$ $P\left(k\right)\Rightarrow P(k+1)$ must be true

If above two conditions are satisfied then only we can say that the statement is true according to the $\text{PMI}.$

In the given question, in the base case, $\text{L.H.S}=1$ and $\text{R.H.S}=4$. Thus, the base case is false and so is the statement.