Two inequality problems 1. Real numbers x and y satisfy the inequality X-Y ≥ 0 and Y ≥ 0 and 2x-y-2 ≥ 0 Find the value range of W = Y-1 / x + 1 2. Sequence 3 / 2, 9 / 4, 25 / 8, 65 / 16161 / 32 N*1/2^n Finding the sum of the first n terms

Two inequality problems 1. Real numbers x and y satisfy the inequality X-Y ≥ 0 and Y ≥ 0 and 2x-y-2 ≥ 0 Find the value range of W = Y-1 / x + 1 2. Sequence 3 / 2, 9 / 4, 25 / 8, 65 / 16161 / 32 N*1/2^n Finding the sum of the first n terms

one
Through the inequality group Y > = 0, X-Y > = 0, 2x-y-2 > = 0, we can carry out linear programming and draw a feasible region. The value range of W = (Y-1) \ (x + 1) can be expressed as the value range of the slope w of the line where the point (x, y) and the point (- 1,1) are located by mathematical language
So, in fact, the meaning of the problem is to find the range of the slope w of the straight line connecting the point (x, y) and the fixed point (- 1,1) in the feasible region
So let the intersection of the line 2x-y-2 = 0 and the X axis be a, that is, a is (1,0), and the fixed point (- 1,1) is B
When the point of feasible region (x, y) is at point a, the slope of line AB is the minimum, that is, the minimum value of W is
w=(0-1)/(1+1)=-1/2
But the slope of the connected line w can only be infinitely close to the slope k = 1 of the line y = x, and can not be equal to or greater than