* Mathematical induction is the method of establishing the truthfulness of a given statement. The statements are generally given in natural numbers. 
* The method follows verification, induction and generalization. A subset of real numbers which satisfy the property of containing the element '1', an element K which belongs to the set such that K+1 also belongs to the set is called an inductive set.
* We observe that the set of natural numbers is the intersection of all the inductive sets, therefore the set of Natural numbers (N) is considered as the smallest inductive set.
 

* An element of a set is least element of the set if no other element of the set is greater than that element i.e. for b ∈ B where B ⊆ N then ∀ x ∈ B b ≤ x, then b is called the least element of B, therefore every non-empty subset of natural numbers has a least element is known as well ordering Principle.
          The mathematical induction can be classified as finite athematical induction and infinite or complete mathematical induction.

Working rule, Finite mathematical induction
¤  The given statement is considered as S(n) where n ∈ N
¤ The statement is proved for n = 1, by substituting n = 1 in R.H.S and the nth term of L.H.S. (Note: If the the nth term is not given, n^th term is calculated and then proved by n = 1)
¤  The statement is assumed to be true for n = K, i.e. replace n with K.
4.  The (K + 1)th term is calculated and is added to both sides of the statement, the R.H.S. is simplified.
¤  The resultant is verified by the given statement by replacing n with K + 1 in R.H.S.
¤  Conclusion is given by– Hence by the Principle of finite mathematical induction the given statement is true for all integral values of n.

In the method of complete mathematical induction
¤  The given statement is considered as S(n) where n ∈ N.
¤  The statement is proved for n = 1 and a series of natural numbers (after finding the nth term).
¤  The statement is assumed to be true for K -1 and K and proving that the statement is true for S (K + 1).
 

Generally the types of problems in mathematical induction are.
             (1)  Identity type                      (2)  Divisibility type.
     In identity type the problems are based on general sequences such as Arithmetic progression (A.P.), Geometric progression (G.P.) and are also based on 
¤  sum of 'n' natural numbers given by  (2a + (n -1) d ) = Sn if they have a common difference 'd' and the first element is 'a' and 'n' gives the number of elements or digits.
                   S_n=   (First term + last term)  (

 (n+1) = S_n) 
 gives sum of first 'n' natural numbers.  
¤  The sum of the squares of first 'n' natural numbers is given by 1² + 2² + ... + n² 
                         
¤  The sum of the cubes of first 'n' natural numbers is given by 1³ + 2³ + ... + n³          
                             
The given sequence is observed and the nth term is calculated.

The n^th term of an A.P. is given by  t_n = a + (n -1)d
The n^th term of a G.P. is given by  t_n = a.r^n-1
Where 'a' is the first element, 'r' is the common ratio and 'n' gives the number of terms.
The sum of 'n' terms of a G.P. is given by 
                         if r > 1
The sum to infinite terms of a G.P. series is given by
     
The problems on divisibility type are based on the concept of division algorithm,
if d1 is divisor, d2 is dividend, or is quotient and r is remainder than
                             (d₂  =  d₁ ×  q + r)
if the remainder is zero, then d1 is a factor of d₂.
¤  In solving the problems, the given statement is considered as S(n).
¤  It is verified for n = 1 to be divisible by the given value or number.
¤  The given statement is assumed to be true for n = k, such that the given statement can be equated to 'm' x the given value. Where m is a constant or a polynomial.
¤  The given statement is proved for n = k + 1
model questions can be referred from different text books.