what is the rationale of multiplying "r" to the second line of series? why does cancelling those terms give us a VALID, sound, logical answer? please help. here's a video of the procedure

i get that it works, but the closest logical explanation i have for myself is 1 - (r^n) divided by 1 - r is the polynomial 1 + r + r^2 + ... + r^(n-1) + r^n. which is identical to the finite geometric series. Thus, distributing (1 - r) to the polynomial gives us 1*S -r*S = S - rS.