From Internet Encyclopedia of Philosophy
The mathematical proof technique called “mathematical induction” is deductive and not inductive. Proofs that make use of mathematical induction typically take the following form:
Property P is true of the natural number 0.
For all natural numbers n, if P holds of n then P also holds of n + 1.
Therefore, P is true of all natural numbers.
When such a proof is given by a mathematician, and when all the premises are true, then the conclusion follows necessarily. Therefore, such an inductive argument is deductive. It is deductively sound, too.