Proving $a^n-b^n=(a-b)\sum_{k=1}^{n}a^{n-k} b^{k-1}$ with induction

Question

How can I prove this using induction. I proved for n=1 but now I'm feeling confused while I'm trying to prove for n+1 because of how the summation develops

$$ a^n-b^n=(a-b)\sum_{k=1}^{n}a^{n-k} b^{k-1} $$

NOTE: I really need to use induction to prove this, that's why the answer of the other question doesn't fit my question...

Answer 1

Write $a^{n+1}-b^{n+1}=(a+b)(a^n-b^n)-ab(a^{n-1}-b^{n-1})$ and then apply the induction hypothesis.

Answer 2

Hint: Write $\;a^{n+1}-b^{n+1}=a(a^n-b^n)+ab^n-b^{n+1}$ and apply the induction hypothesis.

However, the most natural way consists in proving first the formula: $$1-x^n=(1-x)(1+x+\dots+x^{n-1})$$ by a very easy induction (actually it is one of the most illuminating examples of induction when one wants to explain it to beginners).

Then set $x=\dfrac ab$.