# Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck.

The problem is that I can not see the telescoping pattern anywhere for following problem.

Prove that $$a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + ... + ab^{p-2} + b^{p-1})$$ using telescoping propery

Usage of telescoping property is actually a hint to the problem but it actually made my life much harder.

Any ideas?I would be very thankful for swift and quick answer.

-
Is $n=p$? If not: how are they related? –  mathmax May 17 at 21:53

$(a−b)\sum_0^{n-1}a^kb^{n-1-k}=\sum_0^{n-1}a^{k+1}b^{n-1-k}-\sum_0^{n-1}a^kb^{n-k}$
$=\sum_1^{n}a^kb^{n-k}-\sum_0^{n-1}a^kb^{n-k}$ $=a^n-b^n$