Prove the following by using the principle of mathematical induction for all :

Let P(n):

I.        For  n = 1,
        P(1) :

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

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

∴  

              which is true
∴             P (m + 1) is true

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