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+1 vote
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in Principle of Mathematical Induction by (46.2k points)

Prove that : 41n – 14n is divisible by 27.

1 Answer

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Best answer

Let the given statement is P(n) where n ∈ N,

i.e. P(n)= 41n – 14n is divisible by 27

For n = 1, P(1) = 411 – 141 = 41 – 14 = 27

which is divisible by 27

Hence, given statement is true for n = 1, 

i.e., P( 1) is true

Let given statement is true for n = m 

i.e., P(m) is true

Then P(m) = 41m – 14m, is divisible by 27 ……(i)

Now, we have to prove that given statement is true for n = m + 1, 

i.e., P(m + 1) is also true

Now,

Here, there are two terms in R.H.S. 

In first term (41m – 14m) is divisible by 27 [From (i)] 

and second term is also divisible by 27 because 27 is its one factor.

Then sum of both terms is also divisible by 27.

Hence, the principle of mathematical in ducation, 

we can say that the given statement is true for each natural numbers n ∈ N.

Hence Proved.

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