Prove the following using the principle of mathematical induction for all n N

41n – 14n is a multiple of 27.


Let the given statement be P(n), as


P(n):41n - 14n is divisible by 27.


First, we check if it is true for n = 1,


P(1):411 - 141 = 27;


It is true for n = 1.


Now we assume that it is true for some positive integer k, such that


P(k):41k - 14k = 27m where m N.


41k = 14k + 27m ………….(1)


We shall prove that P(k + 1)is true,


P(k + 1):41k + 1 - 14k + 1


41k.41 - 14k + 1


(14k + 27m)41 - 14k + 1 From equation(1)


27.41m + 14k(41 - 14)


27.41m + 14k.27


27(41m + 14k)


We proved that P(k + 1) is true.


Hence by principle of mathematical induction it is true for all n N.


23
1