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If $(1+\sqrt 2)^n=x_n+y_n\sqrt 2$, where $x_n,y_n$ are integers, then

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Given: $(1+\sqrt 2)^n=x_n+y_n\sqrt 2$
$\Rightarrow\:(1-\sqrt 2)^n=x_n-y_n\sqrt 2$
$\Rightarrow\:(1+\sqrt 2)^n.(1-\sqrt 2)^n=(x_n+y_n\sqrt 2).(x_n-y_n\sqrt 2)$
$\Rightarrow\:(1-2)^n=(-1)^n=x_n^2-2y_n^2$
answered Oct 23, 2013