Q. 114.0( 50 Votes )

Prove the following using the principle of mathematical induction for all n N


Answer :

Let the given statement be P(n), as



First, we check if it is true for n = 1,



It is true for n = 1.


Now we assume that it is true for some positive integer k, such that


…………..(1)


We shall prove that P(k + 1)is true,


Solving the left hand side with n = k + 1




[From equation (1)]









Which is equal to the Right hand side for n = k + 1.We proved that P(k + 1) is true.


Hence by principle of mathematical induction it is true for all n N.


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