Question

41^n-14^n is multiple of 27 prove it by multiple induction​

Answer 1

Step-by-step explanation:

Say, 41^n-14^n is multiple of 27 = P(n)

For n = 1, 41¹ - 14¹ = 41 - 14 = 27, P(1) is true.

Let it be true for P(m), means,

Let 41^(m) - 14^(m) is divisible by 27.

Let, 41^m - 14^m = 27A

For n = m + 1,

=> 41^(m+1) - 14^(m+1)

=> 41^(m).41¹ - 14^(m).14¹

=> 41.41^m - 14.14^m

=> 41.41^m - 41.14^m + 27.14^m

=> 41(41^m - 14^m) + 27.14^m

=> 41(27A) + 27(14^m)

=> 27[41A + 14^m]

As 27 is outside the brackets, it is divisible by 27.proved.