Given that \(\frac{1}{\sqrt{y}+\sqrt{z}}\) is the arithmetic mean of \(\frac{1}{\sqrt{x}+\sqrt{z}}\) and \(\frac{1}{\sqrt{x}+\sqrt{y}}\), we have: \(\frac{2}{\sqrt{y}+\sqrt{z}}=\frac{1}{\sqrt{x}+\sqrt{z}}+\frac{1}{\sqrt{x}+\sqrt{y}}\)
\(⇒\)\(2(\sqrt{x} + \sqrt{z})(\sqrt{x} + \sqrt{y}) = (\sqrt{y} + \sqrt{z})(\sqrt{x} + \sqrt{y} + \sqrt{x} + \sqrt{z})\)
\(⇒\)\(2(x + \sqrt{xy} + \sqrt{xz} + \sqrt{yz}) = 2\sqrt{xy} + y + \sqrt{yz} + 2\sqrt{xz} + \sqrt{yz} + z\)
\(⇒\ 2x=y+z\)
This implies that x is the arithmetic mean of y and z, meaning y, x, and z are in an arithmetic progression. The correct option is (A): \(y ,x\) and \(z\) are in arithmetic progression.
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: