# Prove that (x+y)(x^2 +y^2)(x^4 +y^4).......(x^{2^(n-1 )} + y^{2^(n-1 )}) = {x^(2^(n)) -y^(2^(n))}/(x-y)?

Apr 29, 2015

If we multiply both the sides for $x - y$, we will obtain:

at the right ${x}^{2 n} - {y}^{2 n}$, and at the left

(remembering that $\left(a - b\right) \left(a + b\right) = {a}^{2} - {b}^{2}$):

$\left(x - y\right) \left(x + y\right) \left({x}^{2} + {y}^{2}\right) \left({x}^{4} + {y}^{4}\right) \ldots \ldots . \left({x}^{{2}^{n - 1}} + {y}^{{2}^{n - 1}}\right) =$

$= \left({x}^{2} - {y}^{2}\right) \left({x}^{2} + {y}^{2}\right) \left({x}^{4} + {y}^{4}\right) \ldots \ldots . \left({x}^{{2}^{n - 1}} + {y}^{{2}^{n - 1}}\right) =$

$= \left({x}^{4} - {y}^{4}\right) \left({x}^{4} + {y}^{4}\right) \ldots \ldots . \left({x}^{{2}^{n - 1}} + {y}^{{2}^{n - 1}}\right) =$

$= \ldots = \left({x}^{{2}^{n - 1}} - {y}^{{2}^{n - 1}}\right) \left({x}^{{2}^{n - 1}} + {y}^{{2}^{n - 1}}\right) = {x}^{2 n} - {y}^{2 n}$.