Recursion Recursion Recursion

$\begin{aligned} x_{n+2} & = 2 x_{n+1} + 3 y_{n+1} \\ & = 2 \left(2 x_{n} + 3 y_{n}\right) + 3 \left( x_{n} + 2 y_{n}\right) \\ & = 7 x_{n} + 12 y_{n} \\ & = 8 x_{n} + 12 y_{n} - x_n \\ & = 4x_{n+1} - x_n \end{aligned}$

$\Rightarrow x_{n+2} - 4x_{n+1} + x_n = 0$

Therefore, the characteristics equation is as follows:

$\begin{aligned} x_{n+2} - 4x_{n+1} + x_n & = 0 \\ \left(x - \frac{4-\sqrt{12}}{2} \right) \left(x - \frac{4 + \sqrt{12}}{2} \right) & = 0 \\ \left(x - 2 + \sqrt{3} \right) \left(x - 2 - \sqrt{3} \right) & = 0 \end{aligned}$

$\begin{aligned} \Rightarrow x_n & = a(2 + \sqrt{3})^n + b(2 - \sqrt{3})^n \\ x_0 & = \color{#3D99F6} {a + b = 2} \\ x_1 & = a(2 + \sqrt{3}) + b(2 - \sqrt{3}) = 2(a+b) + \sqrt{3}(a-b) \\ & = 4 + \sqrt{3}(a-b) \\ x_1 & = 2x_0 + 3y_0 = 2(2) + 3(1) = 7 \\ \Rightarrow 4 + \sqrt{3}(a-b) & = 7 \\ \sqrt{3}(a-b) & = 3 \\ a - b & = \sqrt{3} \\ \color{#3D99F6} {a + b} & \color{#3D99F6} {= 2} \\ \Rightarrow a & = \frac{2+\sqrt{3}}{2} \\ \Rightarrow b & = 2 - a = 2 - \frac{2+\sqrt{3}}{2} = \frac{2-\sqrt{3}}{2} \\ x_n & = a(2 + \sqrt{3})^n + b(2 - \sqrt{3})^n \\ \Rightarrow x_n & = \frac{(2 - \sqrt{3})^{n+1} + (2 + \sqrt{3})^{n+1}}{2} \end{aligned}$

$\Rightarrow A = 2 \quad \Rightarrow B = 1 \quad \Rightarrow C = 3 \quad \Rightarrow D = 2$

$\begin{aligned} \Rightarrow \sum_{k=B}^A x_k + \sum_{k=D}^C y_k & = \sum_{k=1}^2 x_k + \sum_{k=2}^3 y_k \\ & = x_1 + x_2 + y_2 + y_3 \\ & = 7 + 26 + 15 + 56 \\ & = \boxed{104} \end{aligned}$