Let the set of complex numbers $({\mathrm{x}}_1,{\mathrm{y}}_1),({\mathrm{x}}_2,{\mathrm{y}}_2),({\mathrm{x}}_3,{\mathrm{y}}_3),..........$ denoting the points on the complex plane satisfying  $({\mathrm{x}}_{\mathrm{n}+1},{\mathrm{y}}_{\mathrm{n}+1})=(\sqrt{3}{\mathrm{x}}_{\mathrm{n}}-{\mathrm{y}}_{\mathrm{n}},\sqrt{3}{\mathrm{y}}_{\mathrm{n}}+{\mathrm{x}}_{\mathrm{n}})$ for $\mathrm{n}=1,2,3,.......$ suppose $({\mathrm{x}}_{100},{\mathrm{y}}_{100})=(2,4)$ then the value of ${\mathrm{x}}_1+{\mathrm{y}}_1$ is equal to 

The point $({\mathrm{x}}_1,{\mathrm{y}}_1),({\mathrm{x}}_2,{\mathrm{y}}_2),...({\mathrm{x}}_{\mathrm{n}+1},{\mathrm{y}}_{\mathrm{n}+1})$ 

Given $({\mathrm{x}}_{\mathrm{n}+1}+\mathrm{i}{\mathrm{y}}_{\mathrm{n}+1})={\mathrm{Z}}_{\mathrm{n}+1}\left(\mathrm{say}\right)$

$\begin{array}{l}\therefore {\mathrm{Z}}_{\mathrm{n}+1}=\left(\sqrt{3}{\mathrm{x}}_{\mathrm{n}}-{\mathrm{y}}_{\mathrm{n}}\right)+\mathrm{i}\left(\sqrt{3}{\mathrm{y}}_{\mathrm{n}}+{\mathrm{x}}_{\mathrm{n}}\right)\\ =\sqrt{3}\left({\mathrm{x}}_{\mathrm{n}}+{\mathrm{iy}}_{\mathrm{n}}\right)-{\mathrm{y}}_{\mathrm{n}}+{\mathrm{ix}}_{\mathrm{n}}=\sqrt{3}\left({\mathrm{x}}_{\mathrm{n}}+{\mathrm{iy}}_{\mathrm{n}}\right)+\mathrm{i}\left({\mathrm{x}}_{\mathrm{n}}-\frac{{\mathrm{y}}_{\mathrm{n}}}{\mathrm{i}}\frac{\mathrm{i}}{\mathrm{i}}\right)\\ =\sqrt{3}\left({\mathrm{x}}_{\mathrm{n}}+{\mathrm{iy}}_{\mathrm{n}}\right)+\mathrm{i}\left(\mathrm{x}+{\mathrm{iy}}_{\mathrm{n}}\right)=\sqrt{3}{\mathrm{Z}}_{\mathrm{n}}+{\mathrm{iZ}}_{\mathrm{n}}\\ ={\mathrm{Z}}_{\mathrm{n}}(\sqrt{3}+\mathrm{i})\text{ where }{\mathrm{Z}}_{\mathrm{n}}={\mathrm{x}}_{\mathrm{n}}+{\mathrm{iy}}_{\mathrm{n}}=2{\mathrm{Z}}_{\mathrm{n}}\left(\cos \frac{\mathrm{\pi }}{6}+\mathrm{isin}\frac{\mathrm{\pi }}{6}\right)\\ {\mathrm{Z}}_{\mathrm{n}+1}=2{\mathrm{Z}}_{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{6}\frac{{\mathrm{Z}}_{\mathrm{n}+1}}{{\mathrm{Z}}_{\mathrm{n}}}=2{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{6}}=\mathrm{K}\left(\mathrm{say}\right)}\end{array}$

putting $\mathrm{n}=1,2,3....99,\mathrm{we}\mathrm{get}$

$\frac{{\mathrm{Z}}_2}{{\mathrm{Z}}_1}=\mathrm{K},\frac{{\mathrm{Z}}_3}{{\mathrm{Z}}_2}=\mathrm{K},...\frac{{\mathrm{Z}}_{100}}{{\mathrm{Z}}_{99}}=\mathrm{K}$

Multiplying above all, we get

$\begin{array}{l}\frac{Z_{100}}{Z_{99}}=K^{99}=2^{99}{e}^{i\frac{33\pi }{2}}=2^{99}i\Rightarrow Z_{100}=2^{99}\left(Z_{1}i\right)\\ \Rightarrow Z_1=\frac{Z_{100}}{2^{99}i}=\frac{2+4i}{2^{99}i}\Rightarrow Z_1-2^{-97}-i{2}^{-98}=x_1+i{y}_1\\ \Rightarrow x_1+y_1=2^{-97}-2^{-98}=2^{-98}(2-1)=2^{-98}\end{array}$