Complex City 3

$\begin{aligned} T_n & = \frac{4 \color{#3D99F6}{ \omega ^{4n}} + 3 \color{#3D99F6} {\omega ^{3n}} + 3\omega^{2n}}{5\omega^{2n} + 3 \omega ^{n} + 3} \\ & = \frac{4 \color{#3D99F6}{ (1\dot{} \omega ^{n})} + 3 \color{#3D99F6} {(1)} + 3\omega^{2n}}{5\omega^{2n} + 3 \omega ^{n} + 3} \quad \quad \color{#3D99F6} {[\omega^{3n}=1]} \\ & = \frac{\omega ^{n} + 3 (1 + \omega ^{n} + \omega^{2n})}{2\omega^{2n} + 3 (1 + \omega ^{n} + \omega^{2n})} \quad \quad \color{#3D99F6} {\text{[There are 3 cases.]}} \\ & = \begin{cases} n \equiv 1 \pmod{3}: & \dfrac{\omega + 3 (\color{#3D99F6}{1 + \omega + \omega^{2}})}{2\omega^{2} + 3 (\color{#3D99F6}{1 + \omega + \omega^{2}})} \\ & = \dfrac{\omega + 3 (\color{#3D99F6}{0})}{2\omega^{2} + 3 (\color{#3D99F6}{0})} = \boxed{\frac{1}{2} \omega^2}\quad \quad \color{#3D99F6} {[1 + \omega + \omega^{2}=0]} \\ n \equiv 2 \pmod{3}: & \dfrac{\omega^2 + 3 (1 + \omega^2 + \color{#3D99F6}{\omega^4})}{2\color{#3D99F6}{\omega^4} + 3 (1 + \omega^2 + \color{#3D99F6}{\omega^4})} \\ & = \dfrac{\omega^2 + 3 (1 + \omega^2 + \color{#3D99F6}{\omega})}{2\color{#3D99F6}{\omega} + 3 (1 + \omega^2 + \color{#3D99F6}{\omega})} = \boxed{\frac{1}{2} \omega} \\ n \equiv 0 \pmod{3}: & \dfrac{\omega^0 + 3 (1 + \omega^0 + \omega^0)}{2 \omega^0 + 3 (1 + \omega^0 + \omega^0)} \\ & = \dfrac{1 + 3(3)}{2 + 3(3)} = \boxed{\frac{10}{11}} \end{cases} \end{aligned}$

$\begin{aligned} \Rightarrow \sum_{n=1}^{66} T_n & = \sum_{k=1}^{22} \left( \frac{1}{2} \omega^2 + \frac{1}{2} \omega + \frac{10}{11} \right) \\ & = \sum_{k=1}^{22} \left(- \frac{1}{2} + \frac{10}{11} \right) \\ & = 22 \left( \frac{9}{22} \right) = \boxed{9} \end{aligned}$

${ T }_{ n }=\cfrac { { 4\omega }^{ 4n }+3{ \omega }^{ 3n }+3{ \omega }^{ 2n } }{ { 5\omega }^{ 2n }+3{ \omega }^{ n }+3 } =\cfrac { 3{ \omega }^{ 2n }+4{ \omega }^{ n }+3 }{ { 5\omega }^{ 2n }+3{ \omega }^{ n }+3 } \quad \left( as\quad { \omega }^{ 3n }=1 \right) \\ \\ =\cfrac { { \omega }^{ n }+3\left( { \omega }^{ 2n }+{ \omega }^{ n }+1 \right) }{ { 2\omega }^{ 2n }+3\left( { \omega }^{ 2n }+{ \omega }^{ n }+1 \right) }$

1.If n=3m+1

$\Rightarrow { \omega }^{ n }=\omega \\ \\ \Rightarrow { T }_{ 3m+1 }=\cfrac { \omega +3\left( { \omega }^{ 2 }+{ \omega }+1 \right) }{ { 2\omega }^{ 2 }+3\left( { \omega }^{ 2 }+{ \omega }+1 \right) } \\ \\ =\cfrac { { \omega }^{ 2 } }{ 2 } \quad \left( as\quad { \omega }^{ 3 }=1,{ \omega }^{ 2 }+{ \omega }+1=0 \right)$

2.If n=3m+2

$\Rightarrow { \omega }^{ n }={ \omega }^{ 2 }\\ \\ \Rightarrow { T }_{ 3m+2 }=\cfrac { { \omega }^{ 2 }+3\left( { \omega }^{ 4 }+{ \omega }^{ 2 }+1 \right) }{ { 2\omega }^{ 4 }+3\left( { \omega }^{ 4 }+{ \omega }^{ 2 }+1 \right) } \\ \\ =\cfrac { { \omega } }{ 2 } \quad \left( as\quad { \omega }^{ 3 }=1,{ \omega }^{ 4 }+{ \omega }^{ 2 }+1=0 \right)$

3.If n=3m+3

$\Rightarrow { \omega }^{ n }=1\\ \\ \Rightarrow { T }_{ 3m+3 }=\cfrac { 1+3\left( 1+1+1 \right) }{ 2+3\left( 1+1+1 \right) } =\cfrac { 10 }{ 11 } \\ \\ \therefore \sum _{ n=1 }^{ 66 }{ { T }_{ n }= } \sum _{ m=0 }^{ 22 }{ \left( { T }_{ 3m+1 }+{ T }_{ 3m+2 }+{ T }_{ 3m+3 } \right) } \\ \\ =22\left( \cfrac { { \omega }^{ 2 } }{ 2 } +\cfrac { { \omega } }{ 2 } +\cfrac { 10 }{ 11 } \right) \\ \\ =22\left( \cfrac { { -1 } }{ 2 } +\cfrac { 10 }{ 11 } \right) =22\left( \cfrac { 9 }{ 22 } \right) =9$