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

Question

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

Accepted Answer

#### Step 1: Given condition
The equation states:  
$$ \frac{1}{\sqrt{x} + \sqrt{z}} = \frac{\frac{1}{\sqrt{x} + \sqrt{y}} + \frac{1}{\sqrt{y} + \sqrt{z}}}{2}. $$

Simplify the right-hand side:  
$$ = \frac{\frac{\sqrt{x} + \sqrt{y} + \sqrt{y} + \sqrt{z}}{(\sqrt{x} + \sqrt{y})(\sqrt{y} + \sqrt{z})}}{2}. $$

Combine terms:  
$$ = \frac{\sqrt{x} + 2\sqrt{y} + \sqrt{z}}{2(\sqrt{x} + \sqrt{y})(\sqrt{y} + \sqrt{z})}. $$

#### Step 2: Equating both sides
Equate both sides:  
$$ \frac{1}{\sqrt{x} + \sqrt{z}} = \frac{\sqrt{x} + 2\sqrt{y} + \sqrt{z}}{2(\sqrt{x} + \sqrt{y})(\sqrt{y} + \sqrt{z})}. $$

Multiply through by the denominators:  
$$ 2(\sqrt{x} + \sqrt{y})(\sqrt{y} + \sqrt{z}) = (\sqrt{x} + 2\sqrt{y} + \sqrt{z})(\sqrt{x} + \sqrt{z}). $$

Expand terms and simplify:  
After simplification, it can be shown that:  
$$ x, y, z $$ are in arithmetic progression.

#### Final Answer:
The relationship that always holds true is **$$ x, y, z $$ are in arithmetic progression**.

Answer

Answer

Answer