let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((n |^ 2) * (4 |^ n)) / ((n + 1) * (n + 2)) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2))) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (2 / 3) + ((($1 - 1) * (4 |^ ($1 + 1))) / (3 * ($1 + 2)));
assume A1: for n being Nat holds s . n = ((n |^ 2) * (4 |^ n)) / ((n + 1) * (n + 2)) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2)))
then A2: s . 0 = ((0 |^ 2) * (4 |^ 0)) / ((0 + 1) * (0 + 2))
.= (0 * (4 |^ 0)) / (1 * 2) by NEWTON:11
.= 0 ;
A3: for n being Nat st n >= 1 & S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( n >= 1 & S₁[n] implies S₁[n + 1] )
assume that
n >= 1 and
A4: (Partial_Sums s) . n = (2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2))) ; :: thesis: S₁[n + 1]
n + 2 >= 2 by NAT_1:11;
then A5: n + 2 > 0 by XXREAL_0:2;
n + 3 >= 3 by NAT_1:11;
then A6: n + 3 > 0 by XXREAL_0:2;
(Partial_Sums s) . (n + 1) = ((2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2)))) + (s . (n + 1)) by A4, SERIES_1:def 1
.= ((2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2)))) + ((((n + 1) |^ 2) * (4 |^ (n + 1))) / (((n + 1) + 1) * ((n + 1) + 2))) by A1
.= ((2 / 3) + ((((n - 1) * (4 |^ (n + 1))) * (n + 3)) / ((3 * (n + 2)) * (n + 3)))) + ((((n + 1) |^ 2) * (4 |^ (n + 1))) / ((n + 2) * (n + 3))) by A6, XCMPLX_1:91
.= ((2 / 3) + ((((n - 1) * (4 |^ (n + 1))) * (n + 3)) / ((3 * (n + 2)) * (n + 3)))) + (((((n + 1) |^ 2) * (4 |^ (n + 1))) * 3) / (((n + 2) * (n + 3)) * 3)) by XCMPLX_1:91
.= (2 / 3) + (((((n - 1) * (4 |^ (n + 1))) * (n + 3)) / ((3 * (n + 2)) * (n + 3))) + (((((n + 1) |^ 2) * (4 |^ (n + 1))) * 3) / ((3 * (n + 2)) * (n + 3))))
.= (2 / 3) + (((((n - 1) * (4 |^ (n + 1))) * (n + 3)) + ((((n + 1) |^ 2) * (4 |^ (n + 1))) * 3)) / ((3 * (n + 2)) * (n + 3))) by XCMPLX_1:62
.= (2 / 3) + (((((n - 1) * (n + 3)) + (((n + 1) |^ 2) * 3)) * (4 |^ (n + 1))) / ((3 * (n + 2)) * (n + 3)))
.= (2 / 3) + (((((n - 1) * (n + 3)) + (((n + 1) * (n + 1)) * 3)) * (4 |^ (n + 1))) / ((3 * (n + 2)) * (n + 3))) by WSIERP_1:1
.= (2 / 3) + ((((4 * (4 |^ (n + 1))) * n) * (n + 2)) / ((3 * (n + 3)) * (n + 2)))
.= (2 / 3) + ((((4 |^ (n + 1)) * 4) * n) / (3 * (n + 3))) by A5, XCMPLX_1:91
.= (2 / 3) + ((((n + 1) - 1) * (4 |^ ((n + 1) + 1))) / (3 * ((n + 1) + 2))) by NEWTON:6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . (1 + 0) = ((Partial_Sums s) . 0) + (s . (1 + 0)) by SERIES_1:def 1
.= (s . 0) + (s . 1) by SERIES_1:def 1
.= ((1 |^ 2) * (4 |^ 1)) / ((1 + 1) * (1 + 2)) by A1, A2
.= (1 * (4 |^ 1)) / ((1 + 1) * (1 + 2))
.= (1 * 4) / ((1 + 1) * (1 + 2))
.= (2 / 3) + (((1 - 1) * (4 |^ (1 + 1))) / (3 * (1 + 2))) ;
then A7: S₁[1] ;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A7, A3);
hence for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2))) ; :: thesis: verum