Question

Let $P\left(n\right)$ be a statement such that truth of $P\left(n\right)\Rightarrow$ the truth of $P(n+1)$ for all $nϵN$, then $P\left(n\right)$ is true

• A. $\forall n>1$
• B. $\forall n$
• C. nothing can be said
• D. $\forall n>k$ (k is some fixed positive integer)

Answer 1

Answer: (C)

nothing can be said

Let $P\left(n\right)$ be a statement such that truth of $P\left(n\right)\Rightarrow P(n+1)$ for all $n\in N$

Now from the statement of the Principle of Mathematical Induction,

we know that if a statement is assumed true for $n=k$ and then if it holds true for $n=k+1$ and $n=1$, then the statement is true for all $n$.

So $P\left(n\right)\Rightarrow P(n+1)$ is given so for $P\left(n\right)$ to be true for all $n\in N$, $P\left(1\right)$ should be true.

But nothing has been stated about $P\left(1\right)$. so nothing can be said about the truth of $P\left(n\right)$.