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Mathematics-Online course: Basic Mathematics - Natural Numbers

# Mathematical Induction

Statements with natural numbers as their parameters can be proved by the Principle of Mathematical Induction. If is a statement that depends on , the method of proof consists of the following two steps:

• Base step (also referred to as initial step): Verify that is true.
• Conclusion (also referred to as inductive step): Show that the assumption ,, is true (induction hypothesis) implies the truth of , i.e.

This establishes the truth of for all .

The Principle of Mathematical Induction successively infers the truth of a statement from the previous statement . Therefore, if in the base step is verfied for some rather than , then the statement has only been proved for .

(Authors: Kimmerle/Abele)

The formula for the sum of square numbers,

can be proved by Mathematical Induction.

Base step ():

Conclusion ( :

As indicated, the induction hypothesis has been applied to obtain the third equality.

(Authors: Kimmerle/Abele)