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### Mathematical induction

A complete induction is usually preceeded by an incomplete induction, which serves to guess the formula.

##### Example 1

Calculate the sum *S _{n}* of the third powers of the first

*n*natural numbers, so

Successively this shows

and you get the second powers of the numbers 1, 3, 6, 10, 15 and so on. Of course, you expect that this continous that way. For the following numbers thus applies

because

The check shows that it is true, and we guess that the sum *S _{n}* is the second power thereoff, so

Now comes the step from *n* to *n + *1 and we get

This result also arises if you replace in the original formula for the sum the *n* by *n + *1. In this way the formula

is proven by mathematical induction.