Q. 195.0( 1 Vote )

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

{(41)n – (14)n} is divisible by 27.

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 multiple of 27


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


=


, where m N …(1)


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


Consider ,




= [ Adding and subtracting ]



[ Using 1 ]




, where r = is a natural number


Therefore is divisible of 27


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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