Induction base case ($P(n_0)$)

The starting instance of a mathematical induction proof, established directly before the inductive step is applied. The implication $P(n)\Rightarrow P(n+1)$ alone can be true when every instance is false. For example, ${\sum }_{j=1}^{n}j=n(n+1)/2+C$ with $C\ne 0$ has a formally correct successor calculation but a false base case.