Question

Use mathematical induction to prove that 2^(3n) - 3^n is divisible by 5 for all positive integers.

ThankS!

Answer 1

check for n=1

2^3 - 3^1 = 8-3 = 5

assume for k:

2^(3k) - 3^k = 5m for some m

now plug in k+1

2^(3(k+1)) - 3^(k+1)
= 2^(3k+3) - 3^(k+1)
= 2^3 * 2^(2k) - 3*3^k
= 8*2^(3k) - 3*3^k
= 3*2^(3k) + 5*2^(3k) - 3*3^k
= 3(2^(3k) - 3^k) + 5*2^(3k)
= 3(5m) + 5*2^(3k)
which is a multiple of 5.