Let P(n) be a statement and let P(n) implies | vedclass

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Question

(D) The principle of mathematical induction states that for a statement $P(n)$ to be true for all natural numbers $n$,two conditions must be satisfied

Vedclass · Accepted Answer

Nothing can be said

Vedclass · Answer

(D) The principle of mathematical induction states that for a statement $P(n)$ to be true for all natural numbers $n$,two conditions must be satisfied: $1$. $P(1)$ must be true. $2$. If $P(k)$ is true,then $P(k+1)$ must be true. In the given problem,only the implication $P(n) \implies P(n+1)$ is provided. Without the base case (i.e.,$P(1)$ being true),we cannot conclude that $P(n)$ is true for any $n$. Therefore,nothing can be said about the truth of $P(n)$.

Let $P(n)$ be a statement and let $P(n) \implies P(n + 1)$ for all natural numbers $n$. Then $P(n)$ is true for:

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