let n be Nat; :: thesis: for s being Real_Sequence st ( for n being Nat holds s . n = ((10 |^ n) + (2 * n)) + 1 ) holds
(Partial_Sums s) . n = (((10 |^ (n + 1)) / 9) - (1 / 9)) + ((n + 1) |^ 2)
let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((10 |^ n) + (2 * n)) + 1 ) implies (Partial_Sums s) . n = (((10 |^ (n + 1)) / 9) - (1 / 9)) + ((n + 1) |^ 2) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (((10 |^ ($1 + 1)) / 9) - (1 / 9)) + (($1 + 1) |^ 2);
assume A1: for n being Nat holds s . n = ((10 |^ n) + (2 * n)) + 1 ; :: thesis: (Partial_Sums s) . n = (((10 |^ (n + 1)) / 9) - (1 / 9)) + ((n + 1) |^ 2)
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 = (((10 |^ (n + 1)) / 9) - (1 / 9)) + ((n + 1) |^ 2) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((10 |^ (n + 1)) / 9) - (1 / 9)) + ((n + 1) |^ 2)) + (s . (n + 1)) by SERIES_1:def 1
.= ((((10 |^ (n + 1)) / 9) - (1 / 9)) + ((n + 1) |^ 2)) + (((10 |^ (n + 1)) + (2 * (n + 1))) + 1) by A1
.= ((((10 |^ (n + 1)) * 10) / 9) - (1 / 9)) + ((((n + 1) |^ 2) + (2 * n)) + 3)
.= ((((10 |^ (n + 1)) * 10) / 9) - (1 / 9)) + (((((n |^ 2) + ((2 * n) * 1)) + (1 |^ 2)) + (2 * n)) + 3) by Lm1
.= ((((10 |^ (n + 1)) * 10) / 9) - (1 / 9)) + (((((n |^ 2) + (2 * n)) + 1) + (2 * n)) + 3)
.= ((((10 |^ (n + 1)) * 10) / 9) - (1 / 9)) + (((n |^ 2) + ((2 * n) * 2)) + (2 |^ 2)) by Lm1
.= ((((10 |^ (n + 1)) * 10) / 9) - (1 / 9)) + ((n + 2) |^ 2) by Lm1
.= (((10 |^ ((n + 1) + 1)) / 9) - (1 / 9)) + ((n + 2) |^ 2) by NEWTON:6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= ((10 |^ 0) + (2 * 0)) + 1 by A1
.= ((10 / 9) - (1 / 9)) + 1 by NEWTON:4
.= (((10 |^ (0 + 1)) / 9) - (1 / 9)) + 1
.= (((10 |^ (0 + 1)) / 9) - (1 / 9)) + ((0 + 1) |^ 2) ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A2);
hence (Partial_Sums s) . n = (((10 |^ (n + 1)) / 9) - (1 / 9)) + ((n + 1) |^ 2) ; :: thesis: verum