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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 Sn 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 Sn 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.