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.