question_answer

Which of the following is the correct principle of Mathematical induction?

A)  Let \[P(n)\] be a statement such that n be any integer and \[P(1)\] is true. Also. \[P(m)\] is true for m, any natural number, then \[P(n)\] is true for all  integers n

B)  Let \[P(n)\] be a statement involving natural number n such that \[P(1)\] is true and\[P(m)\]  is true whenever \[P(n)\] is true for every \[n\ge m,\]then \[P(n)\] is true for all \[n\in N\] (set of natural numbers)

C)  Let\[P(n)\] be a statement where \[n\in N\]such that\[P(1)\] is true and \[P(n),\,P(n+1)\]also holds, then \[P(n)\] is true \[\forall n\in N\]

D)  Let \[P(n)\] be a statement involving the natural number n, such trial P(1) is true and s\[P(m+1)\] is true for all \[n\le m\]Then, \[P(n)\] is true for all \[n\in N\]