For any natural number n, Prove that

.

The proof is obtainted by applying principle of mathematical induction.

Let the given statement be

P(n): 

For n = 1, we have

P(1) :

Thus P(1) is true.

Suppose P(k) is true for some positive integer k, i.e.

                               (1)

We shall prove that P(k+1) is also true, i.e.

Now,

= (a+b)  

 

[From (1)]   

                                                                                                       (By actual multiplication)

                                                                                                         [grouping like terms]

    

 Thus, it has been proved that P(k +1) is true whenever P(k) is true.

 By principle of mathematical induction, P(n) is true for every positive integer n.