let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((2 * n) + 1) * (3 |^ n) ) implies for n being Nat holds () . n = (n * (3 |^ (n + 1))) + 1 )
defpred S1[ Nat] means () . \$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 () . 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 () . (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;
() . 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 () . n = (n * (3 |^ (n + 1))) + 1 ; :: thesis: verum