i'm currently studying the dennis freeman's 6.01 from mit ocw. in the lec 3, where the prof. freeman shows a pair of equations for an accumulator that processes signals. this is it. there are channels and the initial state, which is "rest", returns $0$ (the delay boxes). i accidentally painted two boxes to blue, don't mind the colors please. the $ R $ below is the rightshift operator which operates on signals.
equation 1: $$ (1 - R)Y = X$$
equation 2: $$ Y = (1 + R + {R^2} + {R^3} + ...)X $$
the equation 2 is the declaration of the image, i have shown.
now, from the slide it says: these systems are equivalent in the sense that if each is initially at rest, they will produce identical outputs from the same input. and
$$(1 - R)Y_1 = X_1 \iff Y_2 = (1 + R + {R^2} + {R^3} + ...)X_2 $$
proof: assume $ X_2 = X_1: $
$$ Y_2 = (1 + R + {R^2} + {R^3} + ...)X_2 $$
$$ = (1 + R + {R^2} + {R^3} + ...)X_1 $$
$$ = (1 + R + {R^2} + {R^3} + ...)(1 - R)Y_1 $$
$$ = ((1 + R + {R^2} + {R^3} + ...) - (R + R^2 + R^3 + ...))Y_1 $$
$$ = Y_1 $$
it follows that $ Y_2 = Y_1 $.
then he proceeds to evaluate the system with synthetic division. i did quite not understand the idea and calculations behind this, can you try to explain in simple terms please?
you can check the pdf here and the course here. thank you.