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