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The inductive hypothesis is the temporary assumption made during the inductive step that the proposition $P(k)$ is true for an arbitrary integer k. It is used as a premise to derive the truth of $P(k+1)$.

Conditions: Used within the proof of the inductive step.; Assumes $P(k)$ is true for a fixed arbitrary k.; Required to prove $P(k+1)$.

The inductive hypothesis is the temporary assumption made during the inductive step that the proposition $P(k)$ is true for an arbitrary integer k. It is used as a premise to derive the truth of $P(k+1)$.

Conditions: Used within the proof of the inductive step.; Assumes $P(k)$ is true for a fixed arbitrary k.; Required to prove $P(k+1)$.

The Principle of Mathematical Induction states that if a proposition $P(n)$ is true for the base case (usually $n=1$) and if the truth of $P(k)$ implies the truth of $P(k+1)$ for any arbitrary integer k, then $P(n)$ is true for all natural numbers n.

Conditions: $P(1)$ is true.; For any k, $P(k)$ implies $P(k+1)$.; n is a natural number.

The Principle of Mathematical Induction states that if a proposition $P(n)$ is true for the base case (usually $n=1$) and if the truth of $P(k)$ implies the truth of $P(k+1)$ for any arbitrary integer k, then $P(n)$ is true for all natural numbers n.

Conditions: $P(1)$ is true.; For any k, $P(k)$ implies $P(k+1)$.; n is a natural number.