The principle of mathematical induction is based on the fifth Peano Postulate, which states that:

If a statement holds for the positive integer 1, and if, whenever it holds for a positive integer, it also holds for that integer's successor, then the statement holds for all positive integers.

This principle is quite fundamental and is usually taken as a non-provable postulate about the natural numbers. It can be used to prove that a proposition `P`(`n`) is true for all `n` ∈ **N**. So, the principle of mathematical induction is:

If

P(1) is true, and

if the truth ofP(r) implies the truth ofP(r+ 1),

thenP(n) is true for alln,n= 1, 2, ....

Of course, one doesn't have to start with 1; one could start with another integer.

As an example, say that we wanted to prove that the `n`^{th} triangular number, 1 + 2 + 3 + ... + `n` is equal to ½`n`(`n` + 1), we could:

- Show that the proposition is true for
`n`= 1. In this case, ½`n`(`n`+ 1) = 1 when`n`= 1, so it is true. - Assume that the proposition is true for some natural number
`r`, so1 + 2 + 3 + ... +`r`= ½`r`(`r`+ 1) - Based on the assumption in step 2, deduce that the proposition is true for
`r`+ 1. In other words, we want to show that1 + 2 + 3 + ... +If we add`r`+`r`+ 1 = ½(`r`+ 1)(`r`+ 2)`r`+ 1 to both sides of the equation in step 2, we get:1 + 2 + 3 + ... +Expanding the right hand side and then factoring, we get`r`+`r`+ 1 = ½(`r`)(`r`+ 1) + (`r`+ 1)1 + 2 + 3 + ... +Q.E.D.`r`+ (`r`+ 1) = ½`r`² + ½`r`+`r`+ 1

1 + 2 + 3 + ... +`r`+ (`r`+ 1) = ½`r`² +^{3}/_{2}`r`+ 1

1 + 2 + 3 + ... +`r`+ (`r`+ 1) = ½(`r`² + 3`r`+ 2)

1 + 2 + 3 + ... +`r`+ (`r`+ 1) = ½(`r`+ 1)(`r`+ 2)

Mathematical induction should not be confused with inductive reasoning, which is something different.

Sources used (see bibliography page for titles corresponding to numbers): 20, 51.