If:

The proposition (1) holds for n= 1;

(2) If the proposition is true for n=i, it is also true for n=i+ 1.

Then, the conclusion holds for all natural numbers (non-zero).

The second mathematical induction:

If:

The proposition (1) holds for n= 1;

(2) If the proposition is true for n≤i, it is also true for n=i+ 1.

Then, the conclusion holds for all natural numbers (non-zero).