Induction for divisibility: $3\mid 12^n -7^n -4^n -1$ I must use mathematical induction to show that 
$a_{n} = 12^n −7^n −4^n −1$ is divisible by 3 for all positive integers n.

Assume true for $n=k$
$a_{k} = 12^k -7^k -4^k -1$
Prove true for $n=k+1$
$a_{k} = 12^{k+1} -7^{k+1} -4^{k+1} -1$
$ = (12^k)(12) - (7^k)(7) - (4^k)(4) -1$
$ = (12^k)(12) - (7^k)(3+4) - (4^k)(3+1) -1$

I'm not really sure about the last step, as someone just told me to do it. Am I supposed to find the right addends to use and then distribute the exponent terms until I get a multiple of the original $a_{k}$? Because I can't get it to work out evenly, and the -1 at the end gives me trouble. Also, I know that $12^n$ is a multiple of three already, but I don't know how to implement that fact to my advantage. Can I prove that $7^{n}-4^{n}-1$ is also a multiple of three and go from there?

Hint: Instead of the last step, I would do this:
\begin{align*} a_{k+1} &= 12^{k+1} -7^{k+1} -4^{k+1} -1 \\
 &= 12^k \cdot 12 - 7^k \cdot 7 - 4^k \cdot 4 - 4 + 3 \\
 &= 12^k \cdot 8 - 7^k \cdot 3 + 4(12^k-7^k-4^k-1)+3 \end{align*}
