Q61 / 160 · AP EAMCET 2009 · Mathematics

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

  • A
    $-2^{n+1}$
  • B
    $2^{n+1}$
  • Correct answer
    C
    $-(-2)^n$
  • D
    $-2^n$

Solution

$\begin{aligned} & \text { Now, }(1+\sqrt{3} i)^n+(1-\sqrt{3} i)^n \\ &= {\left[2\left(\frac{1+\sqrt{3} i}{2}\right)\right]^n+\left[2\left(\frac{1-\sqrt{3} i}{2}\right)\right]^n } \\ &=\left(-2 \omega^2\right)^n+(-2 \omega)^n \\ &=(-2)^n\left[\left(\omega^2\right)^{3 r+1}+(\omega)^{3 r+1}\right] \\ & {[\because n=3 r+1, \text { where } r \text { is an integer }] } \\ &=(-2)^n\left(\omega^2+\omega\right)=-(-2)^n\end{aligned}$

Hence, the correct answer is option: 'C'.

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