let-put-1-i-3-simlify-A-n-k-0-n-k-

Question Number 29164 by abdo imad last updated on 04/Feb/18

l e t p u t α = 1 + i \sqrt{3} s i m l i f y

A_{n} = \sum_{k = 0}^{n} α^{k} .

Commented by abdo imad last updated on 06/Feb/18

e h a v e α \neq 1 s o A_{n} = \frac{1 - α^{n + 1}}{1 - α} = \frac{1 - {(1 + i \sqrt{3})}^{n + 1}}{- i \sqrt{3}}

= \frac{i}{\sqrt{3}} (1 - {(1 + i \sqrt{3})}^{n + 1} b u t w e h a v e

1 + i \sqrt{3} = 2 (\frac{1}{2} + i \frac{\sqrt{3}}{2}) = 2 e^{i \frac{π}{3}} \Rightarrow {(1 + i \sqrt{3})}^{n} = 2^{n} e^{i \frac{n π}{3}}

\Rightarrow A_{n} = \frac{i}{\sqrt{3}} (1 - 2^{n} e^{i \frac{n π}{3}}) = \frac{i}{\sqrt{3}} (1 - 2^{n} c o s (\frac{n π}{3}) - i 2^{n} s i n (\frac{n π}{3}))

= \frac{2^{n}}{\sqrt{3}} s i n (\frac{n π}{3}) + \frac{i}{\sqrt{3}} (1 - 2^{n} c o s (\frac{n π}{3}) .