autre solution

salut a tous :

je vous présente une autre solution pour exercice 3 du deuxième olympiade :

Citation :: Exercice 3 :
Soient x et alpha deux nombres réels tel que: .
Montrer que pour tout entier naturel n, .

solution :

$autre solution Gif.download?\dpi{120}&space;\fn_cm&space;on\&space;sait\&space;que\&space;\&space;x+\frac{1}{x}=2cos(\alpha)&space;\&space;et&space;\&space;-1\leq&space;cos(\alpha)\leq&space;1&space;\\\\&space;\left\{\begin{matrix}&space;si\&space;x%3E&space;0\&space;\&space;donc\&space;x+\frac{1}{x}\geq&space;2\&space;\&space;&&space;\\\\&space;si\&space;x%3C0\&space;donc\&space;x+\frac{1}{x}\leq&space;-2&space;\&space;\&space;&&space;\end{matrix}\right.\Rightarrow&space;\left\{\begin{matrix}&space;2cos(\alpha)\geq&space;2\Rightarrow&space;cos(\alpha)=1\Rightarrow&space;\alpha=0\&space;\&space;et\&space;\&space;x=1\&space;&&space;\\\\&space;2cos(\alpha)\leq&space;-2\Rightarrow&space;cos(\alpha)=-1&space;\Rightarrow&space;\alpha=\pi\&space;\&space;et\&space;\&space;x=-1&space;&&space;\end{matrix}\right.\\\\\\\left\{\begin{matrix}&space;si\&space;\&space;\alpha=0\&space;\&space;et\&space;\&space;x=1\&space;\&space;donc\&space;2cos\left&space;(n\alpha&space;\right&space;)=2cos(0)=2&space;\&space;\&space;et\&space;\&space;x^n+\frac{1}{x^n}=2\Rightarrow&space;x^n+\frac{1}{x^n}=2cos(n\alpha)\&space;\&space;{\color{Blue}&space;(1)}&space;&&space;\\\\&space;si\&space;\&space;\alpha=\pi\&space;\&space;et\&space;\&space;x=-1\Rightarrow&space;\left\{\begin{matrix}&space;n=2k\&space;\&space;\&space;x^n+\frac{1}{x^n}=2\&space;et\&space;2cos(n\alpha)=2cos(2k\pi)=2\Rightarrow&space;x^n+\frac{1}{x^n}=2cos(n\alpha)=2\&space;\&space;\&space;\&space;&&space;\\\\n=2k+1\&space;\&space;x^n+\frac{1}{x^n}=-2\&space;et\&space;2cos(n\alpha)=cos(\pi)=-2\Rightarrow&space;x^n+\frac{1}{x^n}=2cos(n\alpha)=-2&&space;\end{matrix}\right.{\color{Blue}&space;(2)}&space;&&space;\end{matrix}\right$

$autre solution Gif.download?\dpi{120}&space;\fn_cm&space;d'o%C3%B9\&space;le\&space;r%C3%A9sultat$

j'attend vos remarques pour la vérité du cet solution si vous voulez Very Happy