The first mathematical induction: (1) proves that the proposition holds when n takes the first value n0.

⑵ Suppose that the proposition holds when n=k(k≥n0, k∈N), and then prove that the proposition also holds when n=k+ 1.

Then this proposition applies to all natural numbers n starting from n0.

The second mathematical induction: (1) proves that the proposition holds when n=n0 and n=n0+ 1.

(2) Suppose that the proposition holds when n = k- 1 and n = k (k ≥ n0, k ∈ n), and then prove that the proposition holds when n=k+ 1.

Then this proposition applies to all natural numbers n starting from n0.

The third mathematical induction: (1) proves that the proposition holds when n takes the first value n0.

⑵ Suppose that the proposition holds when n≤k(k≥n0, k∈N), and then prove that the proposition holds when n=k+ 1.

Then this proposition applies to all natural numbers n starting from n0.

Example:

It is proved that an+bn is divisible by a+b (n(N, n is an odd number).

Proof: ① When n= 1, obviously.

② If n=k, the conclusion is correct. Then when n = k+2,

∵ AK (2+BK (2 = AK (2+A2BK-A2BK+BK (2 = A2 (AK+BK)-BK (A-B)) is divisible by A+B according to the inductive hypothesis.

From ① and ②, we can see that all odd numbers N, An+BN can be divisible by A+B.