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)?

1 Answer
Apr 29, 2015

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

at the right x^(2n)-y^(2n), and at the left

(remembering that (a-b)(a+b)=a^2-b^2):

(x-y)(x+y)(x^2 +y^2)(x^4 +y^4).......(x^{2^(n-1 )} + y^{2^(n-1 )})=

=(x^2-y^2)(x^2 +y^2)(x^4 +y^4).......(x^{2^(n-1 )} + y^{2^(n-1 )})=

=(x^4 -y^4)(x^4 +y^4).......(x^{2^(n-1 )} + y^{2^(n-1 )})=

=...=(x^{2^(n-1 )} - y^{2^(n-1 )})(x^{2^(n-1 )} + y^{2^(n-1 )})=x^(2n)-y^(2n).