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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