simplify-A-n-1-1-3-i-n-1-1-3-i-n-

Question Number 128498 by mathmax by abdo last updated on 07/Jan/21

simplify A_{n} = \frac{1}{{(1 + \sqrt{3} i)}^{n}} + \frac{1}{{(1 - \sqrt{3} i)}^{n}}

Answered by TheSupreme last updated on 08/Jan/21

1 + i \sqrt{3} = 2 e^{i \frac{π}{3}}

A_{n} = \frac{1}{2^{n}} e^{- i \frac{n π}{3}} + \frac{1}{2^{n}} e^{i \frac{n π}{3}} =

= \frac{1}{2^{n}} (2 c o s (\frac{π}{3})) = \frac{1}{2^{n}}