Questions mathématiques diverses

Question 99:
Si on a : $$\begin{align*} ❖\;&\alpha=e^{i\frac{2\pi}{7}}\\ ❖\;&f(x)=A_0+\sum\limits_{k=1}^{20}{A_kx^k}\\ ❖\;&\sum\limits_{k=0}^{6}{f\left({\alpha^kx}\right)}=\beta\left({A_0+A_7x^7+A_{14}x^{14}}\right) \end{align*}$$ Déterminer le réel $\beta$

Réponse 99:
$\sum\limits_{k=0}^{6}{f(\alpha^kx)}=7A_0+\sum\limits_{k=1}^{20}{A_kx^k(1+\alpha^k+...+\alpha^{6k})}$
si $k\neq 7$ et $k\neq 14$; $1+\alpha^k+...+\alpha^{6k}=\dfrac{1-\alpha^{7k}}{1-\alpha^k}=0$ car $\alpha^{7k}=1 $donc $$\begin{align*} \sum\limits_{k=0}^{6}{f(\alpha^kx)}&=7A_0+A_7x^7(1+\alpha^7+...+\alpha^{6\times 7})\\ &+A_{14}x^{14}(1+\alpha^{7\times 2}+...+\alpha^{7\times 14})\\ =&7A_0+7A_7x^7+7A_{14}x^{14}\\ =&7\left({A_0+A_7x^7+A_{14}x^{14}}\right) \end{align*}$$