If a, b and c are in Geometric Progression and $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}$ then, x, y, z are in

1. Arithmetic Progression

2. Geometric Progression

3. $\frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

4. $x=y+z$

We are given:

\(a^{\tfrac{1}{x}} = b^{\tfrac{1}{y}} = c^{\tfrac{1}{z}} = k\)

\(\Rightarrow a = k^x,\; b = k^y,\; c = k^z\)

Since \(a, b, c\) are in G.P.:

\(b^2 = ac\)

\(\Rightarrow (k^y)^2 = (k^x)(k^z)\)

\(\Rightarrow k^{2y} = k^{x+z}\)

\(\Rightarrow 2y = x+z\)

This implies \(y\) is the arithmetic mean of \(x\) and \(z\).

x, y, z are in Arithmetic Progression.