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.

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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