In an A.P. with first term 'a' and the common difference `d(a, d!= 0)`, the ratio `'rho'` of the sum of the first `n` terms to sum of `n` terms succedding them does not depend on `n`. Then the ratio `(a)/(d)` and the ratio `'rho'`, respectively are

Options

A. `(1)/(2), (1)/(4)`
B. `2, (1)/(3)`
C. `(1)/(2), (1)/(3)`
D. `(1)/(2), 2`

Answer

To solve the problem, we need to find the ratios \( \frac{a}{d} \) and \( \rho \) given that the ratio of the sum of the first \( n \) terms to the sum of the \( n \) terms succeeding them does not depend on \( n \).

### Step 1: Find the sum of the first \( n \) terms of the A.P.

The sum of the first \( n \) terms \( S_n \) of an arithmetic progression (A.P.) is given by the formula:

\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]

### Step 2: Find the sum of the next \( n \) terms (i.e., the sum of terms from \( n+1 \) to \( 2n \))

The sum of the first \( 2n \) terms \( S_{2n} \) is:

\[ S_{2n} = \frac{2n}{2} \left(2a + (2n-1)d\right) = n \left(2a + (2n-1)d\right) \]

Thus, the sum of the next \( n \) terms (from \( n+1 \) to \( 2n \)) is:

\[ S_{2n} - S_n = n \left(2a + (2n-1)d\right) - \frac{n}{2} \left(2a + (n-1)d\right) \]

### Step 3: Simplify the expression for \( S_{2n} - S_n \)

Calculating \( S_{2n} - S_n \):

\[ S_{2n} - S_n = n \left(2a + (2n-1)d\right) - \frac{n}{2} \left(2a + (n-1)d\right) \]

Expanding both terms:

\[ = n(2a + 2nd - d) - \frac{n}{2}(2a + nd - d) \]

Combining the terms:

\[ = 2an + 2n^2d - nd - \left(an + \frac{n^2d}{2} - \frac{nd}{2}\right) \]

### Step 4: Collect like terms

Combining like terms gives:

\[ = 2an + 2n^2d - nd - an - \frac{n^2d}{2} + \frac{nd}{2} \]

This simplifies to:

\[ = an + \left(2n^2 - \frac{n^2}{2}\right)d + \left(-nd + \frac{nd}{2}\right) \]

### Step 5: Set up the ratio \( \rho \)

Now, we can express \( \rho \) as:

\[ \rho = \frac{S_n}{S_{2n} - S_n} \]

Substituting the expressions we found:

\[ \rho = \frac{\frac{n}{2}(2a + (n-1)d)}{an + \left(2n^2 - \frac{n^2}{2}\right)d + \left(-nd + \frac{nd}{2}\right)} \]

### Step 6: Analyze the condition that \( \rho \) does not depend on \( n \)

For \( \rho \) to not depend on \( n \), the coefficients of \( n \) in the numerator and denominator must cancel out. This leads us to set certain terms equal to zero.

### Step 7: Solve for \( \frac{a}{d} \) and \( \rho \)

From the analysis, we find that:

1. Setting the coefficient of \( n \) in the denominator to zero gives us \( a - \frac{d}{2} = 0 \) or \( a = \frac{d}{2} \).
2. Substituting this back into the expression for \( \rho \) leads to \( \rho = \frac{1}{3} \).

### Final Ratios

Thus, we have:

\[ \frac{a}{d} = \frac{1}{2}, \quad \rho = \frac{1}{3} \]