A more rigorous way to prove this?

Question

I would like to prove the following statement $$x^n-a^n=(x-a)\sum^{n-1}_{k=0}x^ka^{n-k-1},\qquad\forall n\in\Bbb N_0$$ I can easily prove it by induction using polynomial long division or series expansion however I am unsure whether or not these are "rigorous enough". I have thought about this for a while but is there a way to prove this statement in a more rigorous way? I think that the use of ellipses and "do this for the remaining $n$" gives me this uncertainty. Therefore I pose my question as follows:

Is there a rigorous way to prove this statement, or is the use of ellipses rigorous enough?

Thank you.

Answer 1

\begin{align} & (x-a)\sum^{n-1}_{k=0}x^ka^{n-1-k} \\[10pt] = & x \sum^{n-1}_{k=0}x^ka^{n-1-k} - a \sum^{n-1}_{k=0}x^ka^{n-1-k} \\[10pt] = & \sum_{k=0}^{n-1} x^{k+1}a^{n-1-k} - \sum_{k=0}^{n-1} x^k a^{n-1-k+1} \\[10pt] = & \sum_{\ell=1}^n x^\ell a^{n-\ell} - \sum_{k=0}^{n-1} x^k a^{n-k}. \end{align} Here we've let $\ell=k+1$. But after that it does not matter whether the index that runs from $1$ to $n$ is called $\ell$ or $k$, so we have \begin{align} & \sum_{k=1}^n x^k a^{n-k} - \sum_{k=0}^{n-1} x^k a^{n-k} \\[10pt] = & \left(\left( \sum_{k=1}^{n-1} x^k a^{n-k}\right)+ x^n a^0 \right) - \left( x^0 a^{n-0} + \sum_{k=1}^{n-1} x^k a^{n-k} \right) \\[10pt] = & x^n - a^n \end{align}

Answer 2

Using a series expansion is definitely rigorous enough. It's the most straightforward way to demonstrate the equality, using only basic algebra.

Answer 3

Develop and change the index: (almost all the terms are canceled except for $k=0$ and $k=n$):

$$(x-a)\sum^{n-1}_{k=0}x^ka^{n-1-k}=\sum^{n-1}_{k=0}x^{k+1}a^{n-1-k}-\sum^{n-1}_{k=0}x^ka^{n-k}=\sum^{n}_{k=1}x^{k}a^{n-k}-\sum^{n-1}_{k=0}x^ka^{n-k}\\=x^n-a^n$$ but I would say that all the proofs you mentioned are also rigorous.

Answer 4

Yet another way: \begin{eqnarray*} (x-a)\sum_{k=0}^{n-1}x^{k}a^{n-k-1} &=&(x-a)a^{n-1}\sum_{k=0}^{n-1}(\frac{x}{% a})^{k} \\ &=&(x-a)a^{n-1}\frac{1-(\frac{x}{a})^{n}}{1-\frac{x}{a}}=(x-a)\frac{% a^{n}-x^{n}}{a-x}=x^{n}-a^{n} \end{eqnarray*}