Question

The first three terms of an arithmetic progression are 1/2, x, 25. The first three terms of a geometric progression are x+1/4, 32 1/2, and y, where x and y are positive numbers. Find the value of X and the value of Y.

Answer 1

(25 - 1/2)/2 = 49/4
so, x = 49/4

(32 1/2)/(50/4) = y/(32 1/2)
y = 169/2

Answer 2

1/2, x, 25 are in AP
x - 1/2 = 25-x
2x = 51/2
x = 51/4

x+1/4, 32 1/2, and y are in GP, so
(65/2) / (x + 1/4) = y / (65/2)
y(x+1/4) = 4225/4
y( 51/4 + 1/4) = 4225/4
y( 13) = 4225/4
y = 4224/52= 325/4

check:
for AP: 1/2, 51/4, 25
51/4 - 1/2 = 49/4
25 - 51/4 = 49/4 , x is correct

for GP:
x+1/4, 32 1/2 or 13 , 65/2 , 325/4

(65/2) / 13 = 5/2
(325/4) / (65/2) = 5/2 , the y is correct

Answer 3

I calculated the difference, then erroneously assigned that to x.

My bad.