A) \[-{{2}^{n+1}}\]
B) \[{{2}^{n+1}}\]
C) \[-{{(-2)}^{n}}\]
D) \[-{{2}^{n}}\]
Correct Answer: C
Solution :
Now, \[{{(1+\sqrt{3}i)}^{n}}+{{(1-\sqrt{3}\,i)}^{n}}\] \[={{\left[ 2\left( \frac{1+\sqrt{3}\,i}{2} \right) \right]}^{n}}+{{\left[ 2\left( \frac{1-\sqrt{3}\,i}{2} \right) \right]}^{n}}\] \[={{(-2\,{{\omega }^{2}})}^{n}}+{{(-2\,\omega )}^{n}}\] \[={{(-2)}^{n}}[{{({{\omega }^{2}})}^{3r+1}}+{{(\omega )}^{3r+1}}]\] (\[\because \]n = 3r + 1, where r is an integer) \[={{(-2)}^{n}}({{\omega }^{2}}+\omega )\] \[=-{{(-2)}^{n}}\]
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