41ⁿ – 14ⁿ is a multiple of 27.

Let P(n) = 41ⁿ – 14ⁿ
P(1) is a multiple of 27
Let us assume P(k) is a multiple of 27
P(k) = 41^k – 14^k = 27M………..(A)
To Prove P(k+1) is divisible by 27 using the result of ( A )
P(k +1) = 41^k+1 - 14^k+1
             = 41^k .41 – 14^k . 14
             = (27M + 14^k) 41 – 14^k . 14
             = 27M 41 +14^k . 41 – 14^k . 14
             = 27M 41 + 14^k (41 – 14)
             = 27M 41 + 14^k (27)
             = 27(41M + 14^k)
P(k+1) is a multiple of 27.
hence the result.
P(K+1) is true.
By the Principle of mathematical induction, P(n) is true for all values of n where n N
Hence proved