let the harmonic mean of two numbers x and y be 9.find the harmonic mean of the four numbers x,y,2x,2y

Answer 1

If x, y are in HP than 1/x, 1/y are in AP.
AM = (x+y)/2xy.
HM = 1/AM = 2xy/(x+y) = 9.
Similarly, AM = (1/x + 1/y + 1/2x + 1/2y)/4
AM = 3(x+y)/8xy
HM = 4*2xy/3(x+y) = 4/3 * 2xy/(x+y)
HM = 4/3 * 9 = 12