Prove by math induction. For any integer n ≥ 1, 7 n − 2 n is divisible by 5.

Question

I get to (7^k *7) - (2^k * 2). Could i split these as so? : (7^k -2^k) (7-2) I'm pretty sure i cant just want a quick answer and how else do i progress in the problem?

 

Inductive Step: Let k ≥ 1. Assume that 7^k − 2
^k
is divisible by 5. Consider
7^k+1 − 2^k+1 = 7 · 7k − 2 · 2k

This is what I get up to. How do I progress?  

Answer 1

Given : 7^k - 2^k is divisible by 5 for integer n

then 7^k and 2^k are both divisible by 5.

So 7^k = 5*M and 2^k = 5*N for integers M and N

7^(k+1) - 2^(k+1) = 7 * 7^k - 2 * 2^k = 7*5M - 2*5N = 35M - 10N = 5(7M - 2N)

which is clearly divisible by 5