prove by induction that $(a^n - b^n)$ is a multiple of $(a - b)$ for $n \geq 1$ Okay so I just finished a final in discrete mathematics and I could not figure out how to finish this proof:
"Prove by mathematical induction, that $(a^n - b^n)$ is a multiple of $(a - b)$ when $a$ and $b$ are integers and $n \geq 1$.
$\underline{Base case: n = 1}$
$(a^1 - b^1) = (a - b)x$
$x = 1$
$a - b = a - b$
$\underline{Inductive Hypothesis:}$
$n = k$ for some $k$
$(a^k - b^k) = (a - b)x$
$\underline{Inductive Step:}$
$n = k + 1$
$(a^{k + 1} - b^{k + 1}) = (a - b)x_1$
$(a \cdot a^{k}) - (b \cdot b^{k}) = (a - b) x_1$
$a \cdot [(a - b)x + b^k] + b \cdot [(a - b)x - a^k]= (a-b)x_1$
$(a -b)xa + ab^k + (a - b)xb - ba^k = (a - b)x_1$
$(a -b)xa + (a - b)xb - ab^k + ba^k = (a - b)x_1$
$(a -b)xa + (a - b)xb + ba^k - ab^k = (a - b)x_1$
.....
I don't know if I did it right but I couldn't get any further than this.

1)$n=1 : a-b ✓$
Hypothesis: $a^n-b^n$ is a multiple of $(a-b)$.
Step $n+1$:
$a^{n+1}-b^{n+1} =a a^n -b b^n= $
$aa^n-ba^n+ba^n -bb^n=$
$(a-b)a^n +b(a^n -b^n)=$
$(a-b)a^n +bk(a-b)=$
$ (a-b)(a^n +bk).$
(Since by hypothesis: $a^n-b^n =k(a-b)$).
