**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:

1. Something is true for the first case (base case);
2. Whenever that same thing is true for a case, it will be true for the next case (inductive case),

then

* That same thing is true for every case by induction.

In the careful language of mathematics, a proof by induction often proceeds as follows:

* State that the proof will be *by induction over n*. (n is the *induction variable*.)
* Show that the statement is true when n is 1.
* Assume that the statement is true for any natural number n. (This is called the *induction step*.)
	+ Show then that the statement is true for the next number, {\displaystyle n+1}.

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
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### Sum of the first n natural numbers

Prove that for all natural numbers *n*:

{\displaystyle 1+2+3+....+(n-1)+n={\tfrac {1}{2}}n(n+1)}
Proof:

First, the statement can be written as:

{\displaystyle 2\sum \_{k=1}^{n}k=n(n+1)} (for all natural numbers *n)*
By induction on *n*,

First, for *n*=1:

{\displaystyle 2\sum \_{k=1}^{1}k=2(1)=1(1+1)},
so this is true.

Next, assume that for some *n*=*n0* the statement is true. That is,:

{\displaystyle 2\sum \_{k=1}^{n\_{0}}k=n\_{0}(n\_{0}+1)}
Then for *n*=*n0*+1:

{\displaystyle 2\sum \_{k=1}^{{n\_{0}}+1}k}
can be rewritten as

{\displaystyle 2\left(\sum \_{k=1}^{n\_{0}}k+(n\_{0}+1)\right)}
Since {\displaystyle 2\sum \_{k=1}^{n\_{0}}k=n\_{0}(n\_{0}+1),}

{\displaystyle 2\sum \_{k=1}^{n\_{0}+1}k=n\_{0}(n\_{0}+1)+2(n\_{0}+1)=(n\_{0}+1)(n\_{0}+2)}
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 n-sided polygon is {\displaystyle (n-2)180} degrees."

The initial starting value is 3, and the interior angles of a triangle is {\displaystyle (3-2)180} degrees. 
Assume that the interior angles of a n-sided polygon is {\displaystyle (n-2)180} degrees. Add on a triangle which makes the figure a {\displaystyle n+1}-sided polygon, and that increases the count of the angles by 180 degrees {\displaystyle (n-2)180+180=(n+1-2)180} 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
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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 nth degree cousin as follows:

* A {\displaystyle 1}st degree cousin is the child of a parent's sibling.
* A {\displaystyle n+1}st degree cousin is the child of a parent's nth degree cousin.

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

* | is a natural number.
* If n is a natural number, then {\displaystyle n|} is a natural number.
* If {\displaystyle n|=m|} then {\displaystyle n=m}.

One can then define the operations of addition and multiplication and so on by mathematical induction. For example:

* {\displaystyle m+|=m|}
* {\displaystyle m+n|=(m+n)|}

Related pages
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* Mathematical proof
* Proof by contradiction