Mathematics - Complex Number Question with Solution | TestHub

$\mathrm{f}(\mathrm{x})+\mathrm{f}(\alpha \mathrm{x})+\mathrm{f}\left({\alpha }^2\mathrm{x}\right)+\dots ..+\mathrm{f}\left({\alpha }^6\mathrm{x}\right)=7{\mathrm{A}}_0+{\sum }_{\mathrm{k}=1}^{20}{\mathrm{A}}_{\mathrm{k}}{\mathrm{x}}^{\mathrm{k}}\left(1+{\alpha }^{\mathrm{k}}+\dots \dots +{\alpha }^{6\mathrm{k}}\right)$ but when $\mathrm{k}\ne 7$ and $\mathrm{k}\ne 14$, then $1+{\alpha }^{\mathrm{k}}+{\alpha }^{2\mathrm{k}}+\dots .+{\alpha }^{6\mathrm{k}}=0$

Hence

$$\begin{array}{rlr}& f(x)+f(\alpha x)+\dots ..+f\left({\alpha }^{6}x\right)=7A_0+7A_7{x}^7+7A_{14}{x}^{14}=7\left(A_0+A_7{x}^7+A_{14}{x}^{14}\right)& \end{array}$$

$k=7$