[Math] Without using the binomial expansion. Show that $(\sqrt 3 +i)^n + (\sqrt 3 -i)^n$ is real for any positive integer $n$.


I have to show that $(\sqrt 3 +i)^n + (\sqrt 3 -i)^n$ is REAL for any positive integer $n$. My initial thought was to use trial and error using values $1,2,3,\ldots,n$ but that does not seem like a thorough proof. Many thanks in advance!

Best Answer

Hint A complex number $z$ is real if and only if $z=\overline{z}.$

(I'm not sure for the positive part)

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