This setup reminds me of solving the Bellman equations in reinforcement learning. The Bellman equations are also a set of recursive relationships that can be written using operator notation. However, they are usually solved iteratively via a dynamic programming approach, so I wanted to see if we could also apply that approach to the simpler case here. Let's say the variable LiL_iLi represents the light after i bounces/steps. Then:
Initialize L0=0L_0 = 0L0=0
L1=Le+K(L0)=LeL_1 = L_e + K(L_0) = L_eL1=Le+K(L0)=Le
L2=Le+K(L1)=Le+KLeL_2 = L_e + K(L_1) = L_e + KL_eL2=Le+K(L1)=Le+KLe
L3=Le+K(L2)=Le+KLe+K2LeL_3 = L_e + K(L_2) = L_e + KL_e + K^2L_eL3=Le+K(L2)=Le+KLe+K2Le
So it does work- as we keep computing more steps, this solution will match the solution given in the next slide.