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The Principal of Mathematical Induction

Mathematical induction is used in mathematics to prove many statements, in particular it is used to prove statements, theorems, or even formulas that are asserted by all natural numbers. When we say “all” natural numbers it means any natural number that we may possibly come across on. In order to prove by mathematical induction, we must first go over some very important rules.


Rules:

1.) If when a statement, theorem, or formula is true for any natural number \(n=k\) then it is also true for \(n=k+1\) 
and
2.) The statement is true for \(n=1\) then the statement will be true for every natural number \(n\)

To prove a statement by induction, the first step that must be done is to prove part one. According to part one, if this is true then step two is also true. After applying step two we are now implying that the statement will be true for \(n=2,3,4\), and so on.


Looking back at step one, when we say “the statement is true for \(n=k\),” this is called an induction assumption or even an induction hypothesis. Our hypothesis is what we assume when we are trying to prove our statement by induction.

One very important thing to remember is that when working with induction most people will show that if \(P\left(k\right)\) is true then \(P\left(k+1\right)\) is also true because this is the most important part about induction, it is what we want to prove. However, a common mistake that is made is remembering to prove \(P\left(1\right)\) true, if this is not shown then we cannot know for sure that \(P\left(2\right)\), \(P\left(3\right)\), and so on is true.