The sum of three numbers in HP is 37 and the sum of their reciprocals is `(1)/(4)`. Find the numbers.

Explanation

Three number are in H.P. can be taken as
`(1)/(a - b), (1)/(a), (1)/(a + d)`
then `(1)/(a - d) + (1)/(a) + (1)/(a + d) = 37`...(i)
and `a - d + a + a + d = (1)/(4) rArr a = (1)/(12)`
from (i), `(12)/(1 - 12d) + 12 + (12)/(1 + 12d) = 37 rArr (12)/(1 - 12d) + (12)/(1 + 12d) = 25`
`rArr (24)/(1 - 144d^(2)) = 25 " " 1 - 144d^(2) = (24)/(25) rArr d^(2) = (1)/(25 xx 144)`
`:. d = -+ (1)/(60)`
`:. a - d, a, a + d " are " (1)/(15), (1)/(12), (1)/(10) " or " (1)/(10), (1)/(12), (1)/(15)`
Hence, three numbers in H.P. are 15,12, 10 or 10, 12, 15