Mathematical induction

Mathematical induction is a special way of proving a mathematical truth. It can be used to prove that something is true for all the natural numbers (or all positive numbers from a point onwards). The idea is that if:
then
In the careful language of mathematics, a proof by induction often proceeds as follows:
Because it is true for 1, then it is true for 1+1 (=2, by the induction step), then it is true for 2+1 (=3), then it is true for 3+1 (=4), and so on.
Examples of proof by induction.
Sum of the first n natural numbers.
Prove that for all natural numbers "n":
Proof:
First, the statement can be written as:
By induction on "n",
First, for "n"=1:
so this is true.
Next, assume that for some "n"="n0" the statement is true. That is,:
Then for "n"="n0"+1:
can be rewritten as
Since formula_12
Hence the proof is complete by induction.
The sum of the interior angles of a polygon.
Mathematical induction is often stated with the starting value 0 (rather than 1). In fact, it will work just as well with a variety of starting values. Here is an example when the starting value is 3: "The sum of the interior angles of a formula_1-sided polygon is formula_15 degrees."
The initial starting value is 3, and the interior angles of a triangle is formula_16 degrees. 
Assume that the interior angles of a formula_1-sided polygon is formula_15 degrees. Add on a triangle which makes the figure a formula_5-sided polygon, and that increases the count of the angles by 180 degrees formula_20 degrees. Since both the base case and the inductive case are handled, the proof is now complete.
There are a great many mathematical objects for which proofs by mathematical induction works. The technical term is a "well-ordered set".
Inductive definition.
The same idea can work to "define" a set of objects, as well as to prove statements about that set of objects.
For example, we can define formula_1th degree cousin as follows:
There is a set of axioms for the arithmetic of the natural numbers which is based on mathematical induction. This is called "Peano's Axioms". The undefined symbols are | and =.
The axioms are
One can then define the operations of addition and multiplication and so on by mathematical induction. For example: