Mathematical induction instance of proof technique
Mathematical induction is a method for proving that a statement 
  
    
      
        P
        (
        n
        )
      
    
    {\displaystyle P(n)}
   is true for every natural number 
  
    
      
        n
      
    
    {\displaystyle n}
  , that is, that the infinitely many cases 
  
    
      
        P
        (
        0
        )
        ,
        P
        (
        1
        )
        ,
        P
        (
        2
        )
        ,
        P
        (
        3
        )
        ,
        …
      
    
    {\displaystyle P(0),P(1),P(2),P(3),\dots }
    all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).
A proof by induction consists of two cases. The first, the base case, proves the statement for 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
   without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case 
  
    
      
        n
        =
        k
      
    
    {\displaystyle n=k}
  , then it must also hold for the next case 
  
    
      
        n
        =
        k
        +
        1
      
    
    {\displaystyle n=k+1}
  . These two steps establish that the statement holds for every natural number 
  
    
      
        n
      
    
    {\displaystyle n}
  . The base case does not necessarily begin with 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  , but often with 
  
    
      
        n
        =
        1
      
    
    {\displaystyle n=1}
  , and possibly with any fixed natural number 
  
    
      
        n
        =
        N
      
    
    {\displaystyle n=N}
  , establishing the truth of the statement for all natural numbers 
  
    
      
        n
        ≥
        N
      
    
    {\displaystyle n\geq N}
  .
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable 
  
    
      
        n
      
    
    {\displaystyle n}
  , which can take infinitely many values.Infinite descent
The method of infinite descent is a variation of mathematical induction which was used by Pierre de Fermat. It is used to show that some statement Q(n) is false for all natural numbers n. Its traditional form consists of showing that if Q(n) is true for some natural number n, it also holds for some strictly smaller natural number m. Because there are no infinite decreasing sequences of natural numbers, this situation would be impossible, thereby showing (by contradiction) that Q(n) cannot be true for any n.
The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on the statement P(n) defined as "Q(m) is false for all natural numbers m less than or equal to n", it follows that P(n) holds for all n, which means that Q(n) is false for every natural number n.Limited Mathematical Induction
If one wishes to prove that a property P holds for all natural numbers less than or equal to n, proving P satisfies the following conditions suffices:
P holds for 0,
For any natural number x less than n, if P holds for x, then P holds for x + 1Prefix induction
The most common form of proof by mathematical induction requires proving in the induction step that
type of example of manifestation of representation of