Prove $\frac{a^n−b^n}{a−b}=\sum _{k=0} ^{n−1} a^kb^{n−1−k}$ by Induction Let $a\ne b\in\Bbb R$. Show that for all $n\in\mathbb{N}$:
$$\frac{a^n−b^n}{a−b}=\sum _{k=0} ^{n−1} a^kb^{n−1−k}$$
I know using induction proof to prove it, but I can do base case but I am having trouble on proving if $n$ is true then $n+1$ is true.
Please give me some hints on this. I tried to use $n+1 - n$ but I can't let the left side equal to the right side.
My knowledge I’d only induction proof and a little bit of binomial theory.

I think you can prove this by inspection, just multiplying
\begin{align*}
a^n-b^n = (a-b)(b^{n-1}+b^{n-2}a+b^{n-3}a^2+\dots+a^{n-1})
\end{align*}
This appears in a book from Elias Stein, Complex Analysis.
