Answer :

To Prove:



Let us prove this question by principle of mathematical induction (PMI) for all natural numbers


Let P(n):


For n = 1 P(n) is true since , which is true


Assume P(k) is true for some positive integer k , ie,


= …(1)


We will now prove that P(k + 1) is true whenever P( k ) is true


Consider ,



[ Using 1 ]


[Multiplying and dividing by 2 on RHS ]



Now ,



Therefore, P (k + 1) is true whenever P(k) is true


By the principle of mathematical induction, P(n) is true for all natural numbers, ie, N.


Hence proved.


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