let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n = ((n |^ 2) * (4 |^ n)) / ((n + 1) * (n + 2)) ) implies for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = (2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2))) )
assume A1: for n being Element of NAT holds s . n = ((n |^ 2) * (4 |^ n)) / ((n + 1) * (n + 2)) ; :: thesis: for n being Element of 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:16
.= 0 ;
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (2 / 3) + ((($1 - 1) * (4 |^ ($1 + 1))) / (3 * ($1 + 2)));
(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)) by NEWTON:15
.= (1 * 4) / ((1 + 1) * (1 + 2)) by NEWTON:10
.= (2 / 3) + (((1 - 1) * (4 |^ (1 + 1))) / (3 * (1 + 2))) ;
then A3: S₁[1] ;
A4: 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 A5: ( n >= 1 & (Partial_Sums s) . n = (2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2))) ) ; :: thesis: S₁[n + 1]
( n + 2 >= 2 & n + 3 >= 3 ) by NAT_1:11;
then A6: ( n + 2 > 0 & n + 3 > 0 ) by XXREAL_0:2;
n in NAT by ORDINAL1:def 13;
then (Partial_Sums s) . (n + 1) = ((2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2)))) + (s . (n + 1)) by A5, 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_1:92
.= ((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_1:92
.= (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_1:63
.= (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_1:2
.= (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 A6, XCMPLX_1:92
.= (2 / 3) + ((((n + 1) - 1) * (4 |^ ((n + 1) + 1))) / (3 * ((n + 1) + 2))) by NEWTON:11 ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A3, A4);
hence for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = (2 / 3) + (((n - 1) * (4 |^ (n + 1))) / (3 * (n + 2))) ; :: thesis: verum