The principle of mathematical induction is based on the fifth Peano Postulate, which states that:
If a statement holds for the positive integer 1, and if, whenever it holds for a positive integer, it also holds for that integer's successor, then the statement holds for all positive integers.
This property is quite fundamental and is usually taken as a non-provable postulate about the natural numbers. It can be used to prove that a proposition P(n) is true for all n ∈ N. So, the principle of mathematical induction is:
If P(1) is true, and
if the truth of P(r) implies the truth of P(r + 1),
then P(n) is true for all n, n = 1, 2, ....
Of course, one doesn't have to start with 1; one could start with another integer.
As an example, say that we wanted to prove that the nth triangular number, 1 + 2 + 3 + ... + n is equal to ½n(n + 1), we could:
Mathematical induction should not be confused with inductive reasoning, which is something different.
Sources used: Elementary Number Theory, Second Edition by Underwood Dudley.