If $n$ is an integer which leaves remainder one when divided by three, then $(1+\sqrt{3} i)^n+(1-\sqrt{3}…
If $n$ is an integer which leaves remainder one when divided by three, then $(1+\sqrt{3} i)^n+(1-\sqrt{3} i)^n$ equals
- $-2^{n+1}$
- $2^{n+1}$
- $-(-2)^n$
- $-2^n$
Solution
$\begin{aligned} & \text { Now, }(1+\sqrt{3} i)^n+(1-\sqrt{3} i)^n \\ &= {\left[2\left(\frac{1+\sqrt{3} i}{2}\right)\right]^n+\left[2\left(\frac{1-\sqrt{3} i}{2}\right)\right]^n } \\ &=\left(-2 \omega^2\right)^n+(-2 \omega)^n \\ &=(-2)^n\left[\left(\omega^2\right)^{3 r+1}+(\omega)^{3 r+1}\right] \\ & {[\because n=3 r+1, \text { where } r \text { is an integer }] } \\ &=(-2)^n\left(\omega^2+\omega\right)=-(-2)^n\end{aligned}$
Asked in: AP EAMCET 2009
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