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:
a2 - b2 = (a + b)(a - b)
and this one, too, is fairly common:
a3 - b3 = (a2 + ab + b2)(a - b).
it's not too hard to see that:
a4 - b4 = (a2 + b2)(a + b)(a - b) = (a3 + a2b + ab2 + b3)(a - b)
do you see the pattern...? once you are at this point, it's natural to conjecture that:
, which you say you can prove.