theorem :: SERIES_4:15

for s being Real_Sequence st ( for n being Nat holds s . n = (((1 / 2) |^ n) + (2 |^ n)) |^ 2 ) holds
for n being Nat holds (Partial_Sums s) . n = (((- (((1 / 4) |^ n) / 3)) + ((4 |^ (n + 1)) / 3)) + (2 * n)) + 3