Why a^n - b^n = (a-b)[a^(n-1)+a^(n-2)b+a^(n-3)b^2+...+b^(n-1)]?
How does one come up with this identity?
I really can't figure this out... Please help.
well, most people know this one:
a^{2} - b^{2} = (a + b)(a - b)
and this one, too, is fairly common:
a^{3} - b^{3} = (a^{2} + ab + b^{2})(a - b).
it's not too hard to see that:
a^{4} - b^{4} = (a^{2} + b^{2})(a + b)(a - b) = (a^{3} + a^{2}b + ab^{2} + b^{3})(a - b)
do you see the pattern...? once you are at this point, it's natural to conjecture that:
$\displaystyle a^n - b^n = (a - b)\left(\sum_{k = 0}^{n-1} a^{n-k-1}b^k \right)$, which you say you can prove.