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