Consider an A.P. with first term a and common difference d. Let `S_k` denote the sum of the first k terms. If `S_(kx)/S_(x)` is independent of x, then

Options

A. a=2d
B. a=d
C. 2a=d
D. none of these

Explanation

To solve the problem, we need to analyze the given information about the arithmetic progression (A.P.) and the sums of its terms.

### Step-by-step Solution:

1. **Understanding the Sum of the First k Terms of an A.P.:**
The sum \( S_n \) of the first \( n \) terms of an arithmetic progression with first term \( a \) and common difference \( d \) is given by the formula:
\[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \]

2. **Finding \( S_{kx} \) and \( S_x \):**
We need to find \( S_{kx} \) and \( S_x \):
- For \( S_{kx} \):
\[ S_{kx} = \frac{kx}{2} \left( 2a + (kx - 1)d \right) \]
- For \( S_x \):
\[ S_x = \frac{x}{2} \left( 2a + (x - 1)d \right) \]

3. **Setting Up the Ratio \( \frac{S_{kx}}{S_x} \):**
We need to evaluate the ratio:
\[ \frac{S_{kx}}{S_x} = \frac{\frac{kx}{2} \left( 2a + (kx - 1)d \right)}{\frac{x}{2} \left( 2a + (x - 1)d \right)} \]
The \( \frac{1}{2} \) cancels out:
\[ \frac{S_{kx}}{S_x} = \frac{kx \left( 2a + (kx - 1)d \right)}{x \left( 2a + (x - 1)d \right)} \]
The \( x \) in the numerator and denominator cancels out:
\[ \frac{S_{kx}}{S_x} = k \cdot \frac{2a + (kx - 1)d}{2a + (x - 1)d} \]

4. **Condition for Independence from \( x \):**
For \( \frac{S_{kx}}{S_x} \) to be independent of \( x \), the \( x \) terms in the numerator and denominator must cancel out. This occurs when the coefficients of \( x \) in both the numerator and denominator are equal.

Expanding both:
- Numerator: \( 2a + kxd - d \)
- Denominator: \( 2a + xd - d \)

The coefficients of \( x \) give us:
- From the numerator: \( kd \)
- From the denominator: \( d \)

Setting these equal for independence from \( x \):
\[ kd = d \]

5. **Solving for \( d \):**
Assuming \( d \neq 0 \) (since if \( d = 0 \), the A.P. is constant), we can divide both sides by \( d \):
\[ k = 1 \]

6. **Finding the Relationship Between \( a \) and \( d \):**
Now we need to ensure that the constant terms also lead to a valid relationship. Setting the constant terms equal:
\[ 2a - d = 0 \implies 2a = d \]

### Conclusion:
The condition that \( S_{kx}/S_x \) is independent of \( x \) leads us to the conclusion that:
\[ 2a = d \]

Thus, the correct answer is **Option C: \( 2a = d \)**.