\textbf{Exercise 3 (4 points) -- Common to all candidates}

For each of the four following statements, indicate whether it is true or false, by justifying the answer. One point is awarded for each correct answer that is properly justified. An unjustified answer earns no points. An absence of an answer is not penalised.

The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). We consider the complex number $c = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and the points S and T with affixes respectively $c ^ { 2 }$ and $\frac { 1 } { c }$.

\begin{enumerate}
  \item \textbf{Statement 1:} The number $c$ can be written as $c = \frac { 1 } { 4 } ( 1 - \mathrm { i } \sqrt { 3 } )$.
  \item \textbf{Statement 2:} For all natural integer $n$, $c ^ { 3 n }$ is a real number.
  \item \textbf{Statement 3:} The points $\mathrm { O }$, $\mathrm { S }$ and T are collinear.
  \item \textbf{Statement 4:} For all non-zero natural integer $n$,
$$| c | + \left| c ^ { 2 } \right| + \ldots + \left| c ^ { n } \right| = 1 - \left( \frac { 1 } { 2 } \right) ^ { n } .$$
\end{enumerate}