i need help on trying to prove this (a^n)-1 is divisible by a-1 for every natural number n and every integer a > 1
Induction works. Skipping all the formalities ...
Inductive hypothesis: Assume $\displaystyle {\color{blue}a^k - 1}$ is divisible by $\displaystyle a - 1$. It remains to show that $\displaystyle a^{k+1} - 1$ is also divisible by $\displaystyle a-1$.
But note that:
$\displaystyle \begin{aligned}a^{k+1} - 1 & = a^ka^1 - 1 \\ & = a^ka {\color{red} - a + a} - 1 \qquad (\text{We're adding '0'}) \\ & = a({\color{blue}a^k - 1}) + (a-1)\end{aligned} $
And the conclusion follows