Question

Let $P\left(n\right)$ be a statement and $P\left(n\right)=P(n+1)\forall n\in N$, then $P\left(n\right)$ is true for what values of $n$?

• A. For all $n$
• B. For all $n>1$
• C. For all $n>m$ , $m$ being a fixed positive integer
• D. Nothing can be said

Answer 1

The correct option is A For all $n$

Given, $P\left(n\right)=P(n+1)\forall n\in N$
Substituting $n-1$ in place of $n$,
$P(n-1)=P\left(n\right)$
Thus if $P\left(k\right)$ is true for some $k$ $\in$ $N$, then it is true for $k-1$ and $k+1$.
Thus, it is true $\forall k$ $\in$ $N$