Stuck with induction Divisibility

Question

I have seen many on the questions on here about induction divisibilty, but I haven't found any question that covers the doubt that I'm having. The preposition says: "For any integer n $\leq$-3, 8 divides 5⁻⁽ⁿ⁺²⁾+2(3^-(n+3))+1"
These are the steps I've done already

1. I prove that P(-3) is divisible by 8.
2. Then I assume that P(k) is divisible by 8 and write it as: 5^−(k+2)+2(3^-(k+3))+1 = 8m. Since the result is a multiple of 8
3. Then I need to prove P(k+1) is divisible by 8 (step that I'm stuck on). I did the following:
   5^−(k+1+2)+2(3^-(k+1+3))+1
   Then I factored the first two terms to make them similar to P(k) so I can substitute for 8m:
   5⁻¹5^−(k+2)+2(3^-(k+3))3^-1+1

And that's how far I've gotten. If only my exponents were positive I would have finished this proof long ago, but they aren't. I'm beginning to study proofs and I know these questions are really basic, but I've been trying to solve this on my own for 3 hours now and I haven't made any progress.
Thank you in advance!

Answer 1

The preposition says: "For any integer n $\leq$-3, 8 divides 5⁻⁽ⁿ⁺²⁾+2(3^-(n+3))+1"

Let $m = -n$. Then $n \leq -3$ becomes $-n \geq 3 \iff m \geq 3$. Also, $-(n + 2) = -n - 2 = m - 2$. Finally, $-(n + 3) = -n - 3 = m - 3$. This means your proposition can now be equivalently stated as

For any integer $m \ge 3$, prove $8$ divides $5^{m-2} + 2\left(3^{m-3}\right) + 1$.

Next, just follow the basic steps you outlined earlier, which you state you should now not have any problems with since you're dealing with non-negative exponents only.