Given that the constraint condition x − 2Y + 7 ≥ 04x − 3Y − 12 ≤ 0x + 2Y − 3 ≥ 0 is satisfied, then the minimum value of Z = x2 + Y2 is 0______ ．

Given that the constraint condition x − 2Y + 7 ≥ 04x − 3Y − 12 ≤ 0x + 2Y − 3 ≥ 0 is satisfied, then the minimum value of Z = x2 + Y2 is 0______ ．

As shown in the figure, make the feasible region, X2 + Y2 is the square of the distance from the point (x, y) to the origin, so the minimum value is the square of the distance from the origin to the straight line x + 2y-3 = 0, that is (|− 3 | 5) 2 = 95, so the answer is: 95

Sum of a higher one sequence (2n + 1) / [n ^ 2 * (n + 1) ^ 2]
This is the general term for Sn

An=(2n+1)/[n^2*(n+1)^2]
=[1/(n^2)]-[1/(n+1)^2]
A1=(1/1)-[1/(2^2)]
A2=[1/(2^2)-[1/(3^2)]
A3=[1/(3^2)-[1/(4^2)]
……
An-1=[1/(n-1)^2]-[1/n^2]
Sn=A1+A2+A3+A4+…… +An-1+An
=(1/1)-[1/(2^2)]+[1/(2^2)-[1/(3^2)]+[1/(3^2)-[1/(4^2)]+…… +[1/(n-1)^2]-[1/n^2]+[1/(n^2)]-[1/(n+1)^2]
=1-[1/(n+1)^2]
=(n^2+2n)/[(n+1)^2]

Sum of sequence: e ^ (n * (n + 1)), n = 0,1,2,3

Σ e ^ (n * (n + 1)) is not easy to find directly
So first find ㏑∑ e ^ (n (n + 1)) = ∑ n (n + 1) = (1 / 6) n (n + 1) (2n + 1) + (1 / 2) n (n + 1)
So the original formula = e ^ (1 / 6) n (n + 1) (2n + 1) + (1 / 2) n (n + 1)