During the weekends, A delivers newspaper in the complex plane. On Saturday, he being at z and delivers newspaper to houses located at z3, z5, z7, ......., z2017 in that order. On Sunday, he being at 1 and delivers newspaper to houses located at z2, z4, z6, ......., z2016 in that order. Mr. A travels in a straight line between two houses. The distance he must travel from his starting point to the last house is \sqrt{2016} on each of the two days.

Engineering

Mathematics

nth Roots of Unity

Question

During the weekends, A delivers newspaper in the complex plane. On Saturday, he being at z and delivers newspaper to houses located at z³, z⁵, z⁷, ......., z²⁰¹⁷ in that order. On Sunday, he being at 1 and delivers newspaper to houses located at z², z⁴, z⁶, ......., z²⁰¹⁶ in that order. Mr. A travels in a straight line between two houses. The distance he must travel from his starting point to the last house is $\sqrt{2016}$ on each of the two days.

$Re (z^{2}) = \frac{1008}{1009}$

| z | = 1

$Re (z^{2}) = \frac{1007}{1008}$

$| z | = \frac{1}{2}$

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$∵ \sum_{K = 1}^{1008} | z^{2 K + 1} - z^{2 K - 1} |$

$= \sum_{K = 1}^{1008} | z^{2 K} - z^{2 K - 2} | = \sqrt{2016}$

$\Rightarrow | z | = 1 & \sum_{K = 1}^{1008} | z^{2 K} - z^{2 K - 2} | = \sqrt{2016}$

$⇒ {|z}^{2} - 1 | \sum_{K = 1}^{1008} {| z |}^{2 K - 2} = \sqrt{2016}$

$\Rightarrow 1008 | z^{2} - 1 | = \sqrt{2016}$

$∴$ | z | = 1

$∴$ Let z = cosθ + i sinθ

$∴ | \cos 2 q + i \sin 2 θ - 1 | = \frac{\sqrt{2016}}{1008}$

⇒ | –2 sin²θ + i. 2 sinθ cosθ| = $\frac{\sqrt{2}}{\sqrt{1008}}$

$\Rightarrow | sinθ | = \frac{1}{\sqrt{2016}}$

$∵$ Re(z²) = cos 2θ = 1 – 2 sin²θ

$= 1 - \frac{1}{1008} = \frac{1007}{1008}$