Understanding the Relationship Between GP Terms and Exponents

The problem asks us to find the relationship between the exponents $x$, $y$, and $z$ given that $a$, $b$, and $c$ are in Geometric Progression (GP) and satisfy the condition $a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z}$.

Defining Geometric Progression

If three terms $a$, $b$, and $c$ are in Geometric Progression, it means the ratio between consecutive terms is constant. This implies that the square of the middle term is equal to the product of the other two terms:

$b^2 = ac$

Using the Given Exponential Relation

We are given the relation:

$a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z}$

Let's introduce a constant, $k$, such that:

$a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z} = k$

From this, we can express $a$, $b$, and $c$ in terms of $k$:

• $a = k^x$
• $b = k^y$
• $c = k^z$

Substituting into the GP Condition

Now, we substitute these expressions for $a$, $b$, and $c$ into the geometric progression condition $b^2 = ac$:

$(k^y)^2 = (k^x)(k^z)$

Using the rules of exponents ($(p^m)^n = p^{mn}$ and $p^m \cdot p^n = p^{m+n}$), we get:

$k^{2y} = k^{x+z}$

Deriving the Relationship Between x, y, and z

Since the bases are the same (and assuming $k$ is positive and not equal to 1), the exponents must be equal:

$2y = x + z$

Conclusion: Arithmetic Progression

The equation $2y = x + z$ is the defining condition for an Arithmetic Progression (AP). This means that $y$ is the arithmetic mean of $x$ and $z$. Therefore, the sequence $x$, $y$, $z$ forms an Arithmetic Progression.

This matches Option 1.