Let P(n): 1 + 2 + 3 + ......... + n =
I. For n = 1,
    P(1) : 1 = is true.
II.  Suppose the statement is true for n = m,
      i.e. P(m):           ....(i)
III.    For n = m + 1,
        P(m + 1): 1 + 2 + 3 + ........ + (m + 1) =
or  [1 + 2 + 3 + ...... + m] + (m + 1) =
                                       
                                         [From (i), 1 + 2 + 3 + ...... + m = ]
∴        P (m + 1):

  

    which is true

∴    P(m + 1) is true

∴    P(m) is true P(m + 1) is true
Hence, by mathematical induction
P(n) is true for all

Let
I.      For n = 1,
       is true.
II.    Suppose the statement is true for n = m,
   
          i.e.,     ... (i)
III.     For n = m + 1,
       
or     
     From (i),

∴  



       which is true

∴     P(m + 1) is true

∴     P(m) is true P(m + 1) is true.
Hence, by mathematical induction, P(n) is true for all

 Let P(n) :
I.    For n = 1,
      P(1) : 1

∴     P(1) is true.
II.   Let the statement be true for n = m, 

∴     P(m) :      ... (i)

III.  For   n = m + 1,                      
      P(m + 1) :
or   
  From (i),

∴   

which is true.

∴   P(m + 1) is true.

∴   P(m) is true P (m + 1) is true
Hence, by principle of mathematical induction, P(n) is true for all

Let P(n): 
I.    For n = 1,
     P(1) : 
∴    P(1) is true.
II.   Let the statement be true
       for n = m,

∴          .... (i)

III.   For n = m + 1,
       P(m + 1):
or    
        From (i),
      

∴  
 
      which is true.
∴    P(m + 1) is true
∴    P(m) is true P(m + 1) is true
Hence, by the principal of mathematical induction, P(n) is true for all .