Solution pour le challenge 3
Il faut penser à utiliser les nombres complexes ! Posons dans toute la suite :
D’après la formule de Moivre :
Le but du jeu est de calculer la partie imaginaire de .
Or, l’expression qui définit est un fragment de la somme géométrique suivante :
et cette somme vaut :
![Rendered by QuickLaTeX.com z](https://math-os.com/wp-content/ql-cache/quicklatex.com-b8939da89e9838a566e5b833bad46f1f_l3.png)
L’étape suivante consiste à observer que la somme et le produit de et
sont aisément calculables. En effet :
![Rendered by QuickLaTeX.com z^{7}=1](https://math-os.com/wp-content/ql-cache/quicklatex.com-8e77b5ffbb63f11086b9f58c87b3cb2b_l3.png)
![Rendered by QuickLaTeX.com A](https://math-os.com/wp-content/ql-cache/quicklatex.com-f4b0c9cb2845ee5f3a5e6bd2f3d0d618_l3.png)
![Rendered by QuickLaTeX.com B](https://math-os.com/wp-content/ql-cache/quicklatex.com-01939be7ab289b87d36819d286094b9a_l3.png)
![Rendered by QuickLaTeX.com P=X^{2}+X+2](https://math-os.com/wp-content/ql-cache/quicklatex.com-fc918f37f206829262c646e42cd4f130_l3.png)
![Rendered by QuickLaTeX.com P](https://math-os.com/wp-content/ql-cache/quicklatex.com-6679074dadeda754275442cb734b0c3e_l3.png)
![Rendered by QuickLaTeX.com \Delta=-7](https://math-os.com/wp-content/ql-cache/quicklatex.com-45c7bd610077aae6911fc95ae6186f0d_l3.png)
![Rendered by QuickLaTeX.com A](https://math-os.com/wp-content/ql-cache/quicklatex.com-f4b0c9cb2845ee5f3a5e6bd2f3d0d618_l3.png)
![Rendered by QuickLaTeX.com A](https://math-os.com/wp-content/ql-cache/quicklatex.com-f4b0c9cb2845ee5f3a5e6bd2f3d0d618_l3.png)
Or, d’une part :
![Rendered by QuickLaTeX.com \sin](https://math-os.com/wp-content/ql-cache/quicklatex.com-ce8e0830719690cf7ae05abd17b0048d_l3.png)
![Rendered by QuickLaTeX.com {\displaystyle \left[0,\frac{\pi}{2}\right]}](https://math-os.com/wp-content/ql-cache/quicklatex.com-8af4e97ff0f7b6614f969a21ebe71dd7_l3.png)
![Rendered by QuickLaTeX.com A](https://math-os.com/wp-content/ql-cache/quicklatex.com-f4b0c9cb2845ee5f3a5e6bd2f3d0d618_l3.png)
![Rendered by QuickLaTeX.com A=\frac{-1+i\sqrt{7}}{2}](https://math-os.com/wp-content/ql-cache/quicklatex.com-fcb4bb26516737f586a66cca176d4dea_l3.png)
En définitive :
Pour consulter l’énoncé, c’est ici