Need help evaluating the inductive step

Question

I need to evaluate the expression: I have to prove that $7^n-2^n$ is divisible by $5$, for $n \geq 0$;

$P(k) \to 7^k - 2^k = 5r$

$P(k+1) \to 7^{k+1} - 2^{k+1}$

I'm starting like this:

$7^{k+1} - 2^{k+1} = 7^k\cdot7 - 2^k\cdot2$

since $7^k = 5r+2^k$ and $2^k = -5r+7^k$

therefore the right side of the equation above can be written as

$7\cdot(5r+2^k) - 2\cdot(5r-7^k)$

from now on I'm confused how to continue because substituting $2^k$ and $7^k$ seems to not end to the desired result $7^{k+1} + 2^{k+1} = 5t$ where $t$ is some integer.

Answer 1

Hint:

$7^{k+1}-2^{k+1}=7\cdot7^{k}-2\cdot2^{k}=5\cdot7^{k}+2\left(7^{k}-2^{k}\right)$

Answer 2

Hint more simple proof than induction $a^n-b^n=(a-b)(a^{n-1}+...b^{n-1})$ so $(a-b)|(a^n-b^n)$