1997 Paper 3 Q3

Year: 1997
Paper: 3
Question Number: 3

Course: UFM Pure
Section: Complex numbers 2

Difficulty: 1700.0 Banger: 1504.6

complex numbers roots of unity De Moivre geometric proof regular polygon factorization vectors summation

Problem

By considering the solutions of the equation \(z^n-1=0\), or otherwise, show that \[(z-\omega)(z-\omega^2)\dots(z-\omega^{n-1})=1+z+z^2+\dots+z^{n-1},\] where \(z\) is any complex number and \(\omega={\rm e}^{2\pi i/n}\). Let \(A_1,A_2,A_3,\dots,A_n\) be points equally spaced around a circle of radius \(r\) centred at \(O\) (so that they are the vertices of a regular \(n\)-sided polygon). Show that \[\overrightarrow{OA_1}+\overrightarrow{OA_2}+\overrightarrow{OA_3} +\dots+\overrightarrow{OA_n}=\mathbf0.\] Deduce, or prove otherwise, that \[\sum_{k=1}^n|A_1A_k|^2=2r^2n.\]

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1504.6

Banger Comparisons: 4

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Problem source

By considering the solutions of the equation $z^n-1=0$, or
otherwise,  show that
\[(z-\omega)(z-\omega^2)\dots(z-\omega^{n-1})=1+z+z^2+\dots+z^{n-1},\]
where $z$ is any complex number and
$\omega={\rm e}^{2\pi i/n}$.
Let $A_1,A_2,A_3,\dots,A_n$ be points equally
spaced around a circle of radius $r$ centred at
$O$ (so that they are the vertices of a regular $n$-sided
polygon).
Show that
\[\overrightarrow{OA_1}+\overrightarrow{OA_2}+\overrightarrow{OA_3}
+\dots+\overrightarrow{OA_n}=\mathbf0.\]
Deduce, or prove otherwise, that
\[\sum_{k=1}^n|A_1A_k|^2=2r^2n.\]

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