I need to know if the binomial theorem can somehow be applied to: $$\frac{a^n - b^n}{a- b}$$ I've done a bit of research but I still don't know where to begin with this one so I can't offer any pointers. Any suggestions?
Edit: thanks for all replies! The question was provoked by this reddit post, which was related to calculus. The most relevant SE post is this one here.
On
$$\frac{a^n-b^n}{a-b} = \frac{(a-b)\sum_{k=1}^{n} a^{n-k}b^{k-1}}{a-b} = \sum_{k=1}^{n} a^{n-k}b^{k-1}$$
Using the familiar "difference of two powers" factorisation.
On
Perhaps an overkill (compared to other answers), but since the geometric sum was mentioned in the comments: $$\frac{a^n-b^n}{a-b}=\frac{a^n\left(1-\left(\frac{b}{a}\right)^n\right)}{a\left(1-\frac{b}{a}\right)}=a^{n-1}\cdot\frac{1-\left(\frac{b}{a}\right)^n}{1-\frac{b}{a}}=a^{n-1}\sum_{k=0}^{n-1}\left(\frac ba\right)^k$$
Let c = a - b.
Hence, a = b + c.
Hence, $$ \frac{a^n - b^n}{a- b} $$
becomes $$ \frac{(b+c)^n - b^n}{c} $$
The numerator can be expanded using binomial theorem. The first term of the expansion will be $b^n$ which will get cancelled out. All other terms have c as a factor which will cancel out with the denominator.
Hope that helps.