Question

Dage: If x be the A. M between. y and z . y bethe G.M between z and x. prove that z be the H.M between x and y.​

Answer 1

Step-by-step explanation:

Given:

X is the arithmetic mean between Y and Z

i.e. X=(
`(Y+Z)/(2)`) → eq1

Y is the geometric mean between Z and X

i.e.
`Y^(2)`=X*Z → eq2

To prove that,

Z is the Harmonic mean between X and Y

Multiply eq1 by Y on both sides gives;

XY=(
`(Y^(2)+ZY )/(2)`) → eq3

substituting eq2 in eq3

Therefore, 2XY=XZ=ZY

on taking Z common we get

2XY=Z(X+Y)

Z=(
`(2XY)/(X+Y)`)

Hence, Z is the harmonic mean between X and Y