State the second principle of mathematical induction.
Second principle of mathematical induction:
Let P(n) be a given statement involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true (or true for any fixed natural number). This step is known as the Basis step.
(ii) If the statement (called Induction hypothesis) is true for 1
i.e, truth of P(k) implies the truth of P(k + 1). This step is known as the Induction (or Inductive) step.
Then P(n) is true for all natural numbers n.
Note: The second principle of mathematical induction is completely equivalent to the first principle of mathematical induction which states that if the basis step and the inductive step are proven, then P(n) is true for all natural numbers.
But the only difference is in the inductive hypothesis step that we assume not only that the statement holds for $n=k$ but also that it is true for all $1 \leq n \leq k$.
Also, the base can be other natural number as well apart 1 in both the principles.