1. pattern for factoring

I need some help continuing the pattern for limits here is what I have so far:

$a^3-b^3=(a-b)(a^2+ab+b^2)$
$a^4-b^4=(a-b)(a+b)(a^2+b^2)$
$a^5-b^5=?$
$a^6-b^6=?$
$a^7-b^7=?$
$a^8-b^8=?$
$a^9-b^9=?$
$a^{10}-b^{10}=?$

Thanks!

2. Originally Posted by qbkr21
I need some help continuing the pattern for limits here is what I have so far:

Thanks!
In general for $n\geq 2$ we have,
$x^n-y^n=(x-y)(x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1})$
Now there are two cool ways of writing that long expression,
$\sum_{k=1}^{n-1} x^ky^{n-1-k}$
Another way is,
$\sum_{\begin{array}{c}i+j=n-1\\i,j\geq 0 \end{array}}x^iy^j$

The second one means that it is the sum of all possibilities of getting an exponent sum of $n-1$.