Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

For $n∈ N$, determine the real part of $(1 + i\sqrt{3})^{n}$.

I just can't find the regularity within it. Thanks.

share|improve this question
    
I am a little confused by your formula. Is it "one plus i times square root of 3" in the parentheses? –  Code-Guru Jan 8 '13 at 21:43
    
@Code-Guru - this is how I read it too –  Belgi Jan 8 '13 at 21:43
3  
@i_a_n: please make sure you wanted the imaginary part outside of the radical. Also, your accept rate is o% and that could disuade people, so please pay attention to it. Regards –  Amzoti Jan 8 '13 at 21:46
1  
You need to work on your accept rate. 0% after 20 questions is unacceptable. –  Jacob Jan 8 '13 at 22:07
1  
Regularities will come quickly if you compute. Maybe powers of $\frac{1}{2}+\frac{\sqrt{3}}{2}i$ will be more familiar. –  André Nicolas Jan 8 '13 at 22:26
add comment

2 Answers

Hint: Consider polar representation

share|improve this answer
add comment

Use De Moivre's formula \begin{align} \left[ 1+i\cdot \sqrt{3} \right]^n = & 2^n \left[ \frac{1}{2}+i\cdot \frac{\sqrt{3}}{2} \right]^n & \\ = & 2^n\cdot \left[\cos\left(\frac{\pi}{3}\right)+i\cdot \sin\left(\frac{\pi}{3}\right) \right]^n & \\ = & 2^n\cdot \left[\cos\left(n\cdot\frac{\pi}{3}\right)+i\cdot \sin\left(n\cdot\frac{\pi}{3}\right) \right] & \\ \end{align} Let's $n= 3\cdot [2\cdot k]+r$. We have $r\in\{0,1,2,3,4,5\}$ and \begin{align} \left[ 1+i\cdot \sqrt{3} \right]^n = & 2^n\cdot \left[\cos\left(r\cdot\frac{\pi}{3}\right)+i\cdot \sin\left(r\cdot\frac{\pi}{3}\right) \right] \\ \end{align} Then we have, $$ Re\Big(\left[ 1+i\cdot \sqrt{3} \right]^{3(2k)+r}\Big)= \begin{cases} 2^n\cdot \left(+\frac{1}{1} \right) & \mbox{ if } r=0 \\ 2^n\cdot \left(+\frac{1}{2} \right) & \mbox{ if } r=1,5 \\ 2^n\cdot \left(-\frac{1}{2} \right) & \mbox{ if } r=2,4 \\ 2^n\cdot \left(-\frac{1}{1} \right) & \mbox{ if } r=3 \\ \end{cases} $$

share|improve this answer
4  
For questions tagged (homework), try to avoid giving complete answers. See this for more informtation. –  robjohn Jan 8 '13 at 22:26
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.