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) : 1 + 3 + 5 + .................... + (2m - 1) = m²                    ...(i)
III.      For n = m + 1,
         P(m + 1) : 1 + 3 + 5 + .......... + [2 (m+1) - 1] = (m + 1)²
or      1 + 3 + 5 + ........... + (2m - 1) + (2m + 1) = (m + 1)²
        From (i),
        

∴       
   

∴       P (m + 1) is true.

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