theorem :: SERIES_1:23

for a being Real
for s being Real_Sequence st a <> 1 & ( for n being Nat holds s . (n + 1) = a * (s . n) ) holds
for n being Nat holds (Partial_Sums s) . n = ((s . 0) * (1 - (a to_power (n + 1)))) / (1 - a)