/sci/ - Science & Math

[math]\frak On\ Factoring\ The\ Sum\ and\ Differences\ of\ Powers\ [/math]

I have a couple of questions related to the title of this post (1st line).
First, is there a general way to factor [math]a^n+b^n[/math]? I believe I've seen one for when [math]n[/math] is odd but none for when [math]n[/math] is even, why?
How do you prove: [eqn]\begin{split}
a^n-b^n &= \sum_{j=0}^{n-1}a^{n-j-1}b^{j}\\
&= (a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+...+a^{2}b^{n-3}+ab^{n-2}+b^{n-1})
\end{split}
[/eqn]
I got filtered by proof wiki
Finally, why do people separate [math]a^n-b^n[/math] into two different equations for odd and even [math]n[/math]s?

Re-posting here because it's taking a while for my question to be answered in the other thread (can't delete it either unfortunately).

On F𝖆ctoring Th𝖊 Sum 𝖆nd Diff𝖊r𝖊nc𝖊s of Pow𝖊rs

I have a couple of questions related to the title of this post (1st line).
First, is there a general way to factor [math]a^n+b^n[/math]? I believe I've seen one for when [math]n[/math] is odd but none for when [math]n[/math] is even, why?
How do you prove: [eqn]\begin{split}
a^n-b^n &= \sum_{j=0}^{n-1}a^{n-j-1}b^{j}\\
&= (a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+...+a^{2}b^{n-3}+ab^{n-2}+b^{n-1})
\end{split}
[/eqn]
I got filtered by proof wiki
Finally, why do people separate [math]a^n-b^n[/math] into two different equations for odd and even [math]n[/math]s?