For some positive and distinct real numbers x, y and z, if \(\rm \frac{1}{\sqrt y+\sqrt z}\) is the arithmetic mean of \(\rm \frac{1}{\sqrt x+\sqrt z}\ and \frac{1}{\sqrt x+\sqrt y}\) then the relationship which will always hold true, is

The very first step we would like to do here is to remove the under-root term in denominator, hence multiplying both numerator and denominator with 

Now asscording to the given question the equation formed will be:

\(\rm \frac{1}{√ x+√ z}+\rm \frac{1}{√ x+√ y}=\rm \frac{2}{√ y+√ z}\)

Now let's try to factorize the denominator

⇒ \(\rm \frac{√ x-√ z}{x-z}+\rm \frac{√ x-√ y}{x-y}=\frac{2(√ y-√ z)}{y-z}\)

Now lets go by the options and we are assuming d to be the common difference of out AP>

If x,  y,  z is in AP. Then, x - z = 2d, x - y = d, y - z = d  

Putting this in our equation we get 3\(\sqrt{x}\)  + 3\(\sqrt{z}\) = 6\(\sqrt{y}\) which we don't know about.

If y, x, z is in AP. Then, x - z = d, x - y = -d,  y - z = 2d 

So \(\rm \frac{√ x-√ z}{d}+\rm \frac{√ x-√ y}{-d}=\frac{2(√ y-√ z)}{2d}\)

⇒ (√x - √z) - (√x - √y) = (√y - √z)

Hence, y , x, z is in AP.