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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 [math]n[/math]. ([math]n[/math] is the induction variable.)
 Show that the statement is true when [math]n[/math] is 1.
 Assume that the statement is true for any natural number [math]n[/math]. (This is called the induction step.)
 Show then that the statement is true for the next number, [math]n+1[/math].
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.
An example of proof by induction is the following:
Prove that for all natural numbers n:
 [math]1+2+3+....+(n1)+n=\tfrac12 n(n+1)[/math]
Proof:
First, the statement can be written as:
 [math]2\sum_{k=1}^n k=n(n+1)[/math] (for all natural numbers n)
By induction on n,
First, for n=1:
 [math]2\sum_{k=1}^1 k=2(1)=1(1+1)[/math],
so this is true.
Next, assume that for some n=n_{0} the statement is true. That is,:
 [math]2\sum_{k=1}^{n_0} k = n_0(n_0+1)[/math]
Then for n=n_{0}+1:
 [math]2\sum_{k=1}^{{n_0}+1} k[/math]
can be rewritten as
 [math]2\left( \sum_{k=1}^{n_0} k+(n_0+1) \right) [/math]
Since [math]2\sum_{k=1}^{n_0} k = n_0(n_0+1),[/math]
 [math]2\sum_{k=1}^{n_0+1} k = n_0(n_0+1)+2(n_0+1) =(n_0+1)(n_0+2)[/math]
Hence the proof is complete by induction.
Similar proofs
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 [math]n[/math]sided polygon is [math](n2)180[/math] degrees."
The initial starting value is 3, and the interior angles of a triangle is [math](32)180[/math] degrees. Assume that the interior angles of a [math]n[/math]sided polygon is [math](n2)180[/math] degrees. Add on a triangle which makes the figure a [math]n+1[/math]sided polygon, and that increases the count of the angles by 180 degrees [math](n2)180+180=(n+12)180[/math] 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 wellordered 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 [math]n[/math]th degree cousin as follows:
 A [math]1[/math]st degree cousin is the child of a parent's sibling.
 A [math]n+1[/math]st degree cousin is the child of a parent's [math]n[/math]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
  is a natural number.
 If [math]n[/math] is a natural number, then [math]n[/math] is a natural number.
 If [math]n = m[/math] then [math]n = m[/math].
One can then define the operations of addition and multiplication and so on by mathematical induction. For example:
 [math]m +  = m[/math]
 [math]m + n = (m + n)[/math]
Related pages
References
 ↑ "The Definitive Glossary of Higher Mathematical Jargon" (in enUS). 20190801. https://mathvault.ca/mathglossary/.
 ↑ "3.4: Mathematical Induction  An Introduction" (in en). 20180425. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Book%3A_A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/03%3A_Proof_Techniques/3.04%3A_Mathematical_Induction__An_Introduction.
 ↑ "Induction". http://discrete.openmathbooks.org/dmoi2/sec_seqinduction.html.

