Harmonic Progression

To use a shortcut, (One may call it cheating), only one of the options are in AP. So only their reciprocals will form an HP. :)

LET ME TELL U THE EASIEST WAY TO ANSWER THIS QUESTION . 1/a-d + 1/a + 1/a+d = 37 ...(1) and a-d + a + a+d = 1/4 ......(2) from (2)
u will get a =1/12

   now , put it in equation  (1) 
  d=1/60 
      and then find the series by putting the value of a and d in equation (1)

=> which is 10 12 15 { brain is a mystery, and i m on the way to solve it }

Let the HP be: 1/x , 1/(x+d), 1/(x+2d)

and

x+x+d+x+2d is the reciprocal #'s. then,

1/x + 1/(x+d) + 1/(x+2d) = 37

3x+3d = 1/4 ----> x+d = 1/12

Then,

1/x + 1/(x+d) + 1/(x+2d) ---> 1/(1/12 - d) + 1/(1/12) + 1/(1/12 + d) = 37

1/[(1-12d)/12] + 1/(1/12) + 1/[(1+12d)/12] = 37

12/(1-12d) + 12 + 12/(1+12d) = 37

12(1+12d) + 12(1-2d) = (37-12)(1-144d^2)

24 = 25(1-144d^2)

1 = (25)(144d^2)

1/(25*144) = d^2

d = +/- 1/60

Then,

x+d = 1/12

Case 1: if d = 1/60,

x+1/60 = 1/12

x = 1/12 - 1/60 = 1/15

x+d = 1/12

x+2d=1/10

Hence, the #'s are 10,12,15.

Case 2: if d = -1/60.

x-1/60 = 1/12

x = 1/10

x+d = 1/12

x+2d = 1/15

Hence, the #'s are 10,12,15.

Then, (Case 1) U (Case 2) = 10,12,15.

Final Answer: 10,12,15.