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 S_{1}[ 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 S_{1}[n] holds

S_{1}[n + 1]

.= ((2 * 0) + 1) * (3 |^ 0) by A1

.= (0 * (3 |^ (0 + 1))) + 1 by NEWTON:4 ;

then A3: S_{1}[ 0 ]
;

for n being Nat holds S_{1}[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

defpred S

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 S

S

proof

(Partial_Sums s) . 0 =
s . 0
by SERIES_1:def 1
let n be Nat; :: thesis: ( S_{1}[n] implies S_{1}[n + 1] )

assume (Partial_Sums s) . n = (n * (3 |^ (n + 1))) + 1 ; :: thesis: S_{1}[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 S_{1}[n + 1]
; :: thesis: verum

end;assume (Partial_Sums s) . n = (n * (3 |^ (n + 1))) + 1 ; :: thesis: S

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 S

.= ((2 * 0) + 1) * (3 |^ 0) by A1

.= (0 * (3 |^ (0 + 1))) + 1 by NEWTON:4 ;

then A3: S

for n being Nat holds S

hence for n being Nat holds (Partial_Sums s) . n = (n * (3 |^ (n + 1))) + 1 ; :: thesis: verum