If, for positive real numbers x, y, z, the numbers `x+y,` 2y and y + z are in harmonic progression, then which one of the following is correct ?

Options

A. x, y, z are in geometric progression
B. x, y, z are in arithmetic progression
C. x, y, z are in harmonic progression
D. None of the above

Answer

A

Explanation

As given :
`x+y,2y" and "y+z` are in harmonic progression.
`2y=((x+y)(y+z))/(x+y+y+z)`
`implies" "(1)/(2y)=(1)/(x+y)+(1)/(y+z)`
`implies" "y(x+2y+z)=(xy+xz+y^(2)+yz)`
`implies" "xy+2y^(2)+yz=xy+xz+y^(2)+yz`
`implies" "y^(2)=xz`
`implies" "x,y,z` are in geometric progession.