[Math] Help proving $9^n-8n-1$ is divisible by $8$ for all $n > 1$ by induction

Question

I have been trying to prove that $9^n-8n-1$ is divisible by $8$ for all $n$ integers greater than 1. My progress:
Let $n = 2$. This gives us the expression equal to $64$ which is a factor of 8. Now assume it is true for $n=k$ . for $n = k+1$ :

$$ 9^{k+1} – 8(k+1) – 1$$
$$ = (8+1)^{k} \times (8+1) -8k – 8 -1 $$

I keep getting stuck on this part. Can someone please hint me how I can proceed by using INDUCTION only?

Answer 1

$9^{k+1} - 8(k+1) - 1 = 9(9^k - 8k - 1) + (64k + 8)$

See what to do now?