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)#?

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Answer 1 · 2015-04-29T08:17:43

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)#.