The sum of three numbers in H.P.is 37 and the sum of their reciprocals is 1/4.Find the numbers.

H.P.means Harmonic Progression

Their reciprocals are in AP

let the numbers in AP be (a - d) , a , (a + d)
sum = 3a = 1/4 => a = 1/12

sum of numbers in HP is
[ 1 / (a - d) ] + (1/a) + [1 / (a + d) ] = 37
[ a^2 + ad + a^2 - d^2 + a^2 - ad ] / [ a (a^2 - d^2) ] = 37
3a^2 - d^2 = 37a^3 - 37ad^2

putting a = 1/12 ,
(1/48) - d^2 = (37/1728) - (37/12) d^2
(25/12)d^2 = 1 / 1728
d^2 = 1 / ( 144 * 25)
d = ± 1 / 60

so the numbers are 15 , 12 , 10