let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (2 |^ n) + ((1 / 2) |^ n) ) implies for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) - ((1 / 2) |^ n)) + 1 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((2 |^ ($1 + 1)) - ((1 / 2) |^ $1)) + 1;
assume A1: for n being Nat holds s . n = (2 |^ n) + ((1 / 2) |^ n) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) - ((1 / 2) |^ n)) + 1
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume (Partial_Sums s) . n = ((2 |^ (n + 1)) - ((1 / 2) |^ n)) + 1 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((2 |^ (n + 1)) - ((1 / 2) |^ n)) + 1) + (s . (n + 1)) by SERIES_1:def 1
.= (((2 |^ (n + 1)) - ((1 / 2) |^ n)) + 1) + ((2 |^ (n + 1)) + ((1 / 2) |^ (n + 1))) by A1
.= ((((- ((1 / 2) |^ n)) + 1) + ((1 / 2) |^ (n + 1))) + (2 |^ (n + 1))) + (2 |^ (n + 1))
.= ((((- ((1 / 2) |^ n)) + 1) + (((1 / 2) |^ n) * (1 / 2))) + (2 |^ (n + 1))) + (2 |^ (n + 1)) by NEWTON:6
.= (((- (((1 / 2) |^ n) * (1 / 2))) + 1) + (2 |^ (n + 1))) + (2 |^ (n + 1))
.= (((- ((1 / 2) |^ (n + 1))) + 1) + (2 |^ (n + 1))) + (2 |^ (n + 1)) by NEWTON:6
.= ((- ((1 / 2) |^ (n + 1))) + 1) + ((2 |^ (n + 1)) * 2)
.= ((- ((1 / 2) |^ (n + 1))) + 1) + (2 |^ ((n + 1) + 1)) by NEWTON:6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (2 |^ 0) + ((1 / 2) |^ 0) by A1
.= 1 + ((1 / 2) |^ 0) by NEWTON:4
.= (2 - 1) + 1 by NEWTON:4
.= ((2 |^ (0 + 1)) - 1) + 1
.= ((2 |^ (0 + 1)) - ((1 / 2) |^ 0)) + 1 by NEWTON:4 ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A2);
hence for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) - ((1 / 2) |^ n)) + 1 ; :: thesis: verum