If a,b,c,d are in GP and `a^x=b^y=c^z=d^u`, then `x ,y,z,u ` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

To solve the problem step by step, we will analyze the given information and derive the required conclusion.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that \( a, b, c, d \) are in a geometric progression (GP). This means that there exists a common ratio \( r \) such that:
\[ b = ar, \quad c = ar^2, \quad d = ar^3 \]

2. **Setting Up the Equation**:
We are given that:
\[ a^x = b^y = c^z = d^u \]
Let's denote this common value as \( k \). Thus, we have:
\[ a^x = k, \quad b^y = k, \quad c^z = k, \quad d^u = k \]

3. **Taking Logarithms**:
Taking the logarithm of each equation, we get:
\[ x \log a = \log k, \quad y \log b = \log k, \quad z \log c = \log k, \quad u \log d = \log k \]

4. **Substituting Values**:
Now substituting the values of \( b, c, d \) in terms of \( a \) and \( r \):
- For \( b \):
\[ y \log b = y \log(ar) = y (\log a + \log r) \]
- For \( c \):
\[ z \log c = z \log(ar^2) = z (\log a + 2\log r) \]
- For \( d \):
\[ u \log d = u \log(ar^3) = u (\log a + 3\log r) \]

5. **Equating the Logarithmic Expressions**:
From the expressions we derived, we can write:
\[ x \log a = y (\log a + \log r) = z (\log a + 2\log r) = u (\log a + 3\log r) \]

6. **Letting \( k = \log k \)**:
Let’s denote \( k = \log k \). Thus, we can express \( x, y, z, u \) in terms of \( k \):
- From \( x \log a = k \):
\[ x = \frac{k}{\log a} \]
- From \( y \log b = k \):
\[ y = \frac{k}{\log a + \log r} \]
- From \( z \log c = k \):
\[ z = \frac{k}{\log a + 2\log r} \]
- From \( u \log d = k \):
\[ u = \frac{k}{\log a + 3\log r} \]

7. **Identifying the Pattern**:
The denominators of \( x, y, z, u \) are:
\[ \log a, \quad \log a + \log r, \quad \log a + 2\log r, \quad \log a + 3\log r \]
These terms are in arithmetic progression (AP) with the first term \( \log a \) and a common difference of \( \log r \).

8. **Conclusion**:
Since \( x, y, z, u \) are inversely proportional to these terms, it follows that \( x, y, z, u \) are in harmonic progression (HP).

### Final Answer:
Thus, \( x, y, z, u \) are in HP.

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