let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((2 * n) + 1) * (3 |^ n) ) implies for n being Nat holds (Partial_Sums s) . n = (n * (3 |^ (n + 1))) + 1 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ($1 * (3 |^ ($1 + 1))) + 1;
assume A1: for n being Nat holds s . n = ((2 * n) + 1) * (3 |^ n) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (n * (3 |^ (n + 1))) + 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 = (n * (3 |^ (n + 1))) + 1 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((n * (3 |^ (n + 1))) + 1) + (s . (n + 1)) by SERIES_1:def 1
.= ((n * (3 |^ (n + 1))) + 1) + (((2 * (n + 1)) + 1) * (3 |^ (n + 1))) by A1
.= ((n + 1) * ((3 |^ (n + 1)) * 3)) + 1
.= ((n + 1) * (3 |^ ((n + 1) + 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) * (3 |^ 0) by A1
.= (0 * (3 |^ (0 + 1))) + 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 = (n * (3 |^ (n + 1))) + 1 ; :: thesis: verum