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).^[1]^[2] The idea is that if:

Something is true for the first case (base case);
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.^[3]

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 $n$ $n$ . (This is called the induction step.)
- Show then that the statement is true for the next number, $n+1$ .

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

Sum of the first n natural numbers

Prove that for all natural numbers n:

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

Proof:

First, the statement can be written as:

2\sum _{k=1}^{n}k=n(n+1)

(for all natural numbers n)

By induction on n,

First, for n=1:

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,:

2\sum _{k=1}^{n_{0}}k=n_{0}(n_{0}+1)

Then for n=n₀+1:

2\sum _{k=1}^{{n_{0}}+1}k

can be rewritten as

2\left(\sum _{k=1}^{n_{0}}k+(n_{0}+1)\right)

Since $2\sum _{k=1}^{n_{0}}k=n_{0}(n_{0}+1),$

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 $(n-2)180$ degrees."

The initial starting value is 3, and the interior angles of a triangle is $(3-2)180$ degrees. Assume that the interior angles of a $n$ -sided polygon is $(n-2)180$ degrees. Add on a triangle which makes the figure a $n+1$ -sided polygon, and that increases the count of the angles by 180 degrees $(n-2)180+180=(n+1-2)180$ 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

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 $n$ th degree cousin as follows:

A $1$ st degree cousin is the child of a parent's sibling.
A $n+1$ st degree cousin is the child of a parent's $n$ th 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