Binomial Theorem

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 \(n\)

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 \(n \geq m\), then \(P(n)\) is true for all \(n \in N\) (set of natural numbers)

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 \(\forall n \in N\)

Let \(P(n)\) be a statement involving the natural number \(n\), such that \(P(1)\) is true and \(P(m+1)\) is true for all \(n \leq m\). Then, \(P(n)\) is true for all \(n \in N\)