Q. 94.0( 5 Votes )

Using the principle of mathematical induction, prove each of the following for all n ϵ N:



Answer :

To Prove:



Let us prove this question by principle of mathematical induction (PMI)


Let P(n):


For n = 1


LHS =


RHS = 1


Hence, LHS = RHS


P(n) is true for n = 1


Assume P(k) is true


……(1)


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


RHS =


LHS =


[ Writing the last


Second term ]


= [From 1]



{ 1 + 2 + 3 + 4 + … + n = [n(n + 1)]/2 put n = k + 1 }




=


[ Taking LCM and simplifying ]


=


= RHS


Therefore ,


LHS = RHS


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


By the principle of mathematical induction, P(n) is true for×


where n is a natural number


Put k = n - 1



Hence proved


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