The statement is true for n = 1
                     Assume that the given statement is true for n = k
        
                       Adding  (k + 1)^th  term on both sides, that is

  we get                      

                                               The statement is true for n = k + 1.
      Hence by the principle of finite mathematical induction the given statement is true for all integral values of 'n'.

        Hence the statement is true for n = 1
        Assume that the statement is true for n = k
     
               Adding (k + 1) ⁿ term that is  

 term as both sides.
        

        Hence by the principle of finite mathematical induction the given statement is true for all integral values of 'n'.
 

3.   49ⁿ + 16 n - 1 is divisible by 64
Sol :   Let n = 1  ;  49 + 16 - 1 = 64 is divisible by 64.
                 Assume that the given statement is true for n = k
                              49^k + 16k - 1 = 64 m where m is any constant
                              49^k  =  64 m - 16 k + 1
                                   To prove for n = k + 1
                             =  49^{k+ 1} + 16 (k + 1) - 1 = 49^k. 49 + 16k + 16 - 1
                             =  (64 m - 16k + 1) 49 + 16k + 15
                             =  64 m × 49 - 16k × 49 + 49 + 16k + 15
                             =  64m × 49 - 16k (49 - 1) + 64
                             =  64m × 49 - 16k (48) + 64
 =  64 m × 49 - 16k(4 × 12) + 64
                                =  64 m × 49 - 64k × 12 + 64
                                =  64 (49m - 12k + 1)
                       The given statement is true for n = k + 1
    Hence by the principle of finite mathematical induction the given statement is true for all integral values of 'n'.
 
 They are in A.P.  ;    To prove for n = 1


                                   

 L.H.S. = R.H.S.
The statement is true for n = 1
 Assume that the given statement is true for n = k
            
     Adding (k + 1)^th term on both sides,    is the term to be added.
               
    
  The given statement is true for n = k + 1.
          Hence by the principle of finite mathematical induction the given statement is true for all integral values of 'n'.
 

5.  Show that 1.2.3 + 2.3.4 + ....  up to 'n' terms =  
                1, 2, 3, ...   t_n = a + (n - 1) d = 1 + (n - 1) .1 = n
                2, 3, 4, ...   t_n = a + (n - 1) d = 2 + (n - 1) .1 = n + 1
                3, 4, 5, ...   t_n = a + (n - 1) d = 3 + (n - 1) .1 = n + 2 are in A.P.
              S(n)  =  1.2.3 + 2.3.4 + ..... + n(n + 1) (n + 2) =  
                                             To prove for n = 1
                         L.H.S. 1 (1 + 1)  (1 + 2)  =  6
  
            L.H.S. = R.H.S.
                   

 The given statement is true for n = 1.
               Assume that the given statement is true for n = k
    S(k)  =  1.2.3. + 2.3.4 +  ....  + k(k + 1) (k + 2)  =  
            Adding (k + 1)th term, that is (k + 1) (k + 2) (k + 3) on both sides.
          1.2.3 + 2.3.4 +  .... +  k(k + 1) (k + 2) + (k + 1) (k + 2) (k + 3)      
                                               
                                    The given statement is true for n = k + 1
     Hence by the principle of finite mathematical induction the given statement is true for all integral values of n.