If `S_(n)` be the sum of first n terms of an A.P. and if `(S_(pn))/(S_(n))` is independent of n then `(a)/(d)` is equal to

Options

A. 2
B. `1//2`
C. 3
D. none

Answer

To solve the problem, we need to analyze the sum of the first n terms of an arithmetic progression (A.P.) and determine the ratio \( \frac{a}{d} \) given that \( \frac{S_{pn}}{S_n} \) is independent of \( n \).

### Step-by-Step Solution:

1. **Formula for the Sum of the First n Terms of an A.P.:**
The sum of the first \( n \) terms \( S_n \) of an A.P. can be expressed as:
\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Calculate \( S_{pn} \):**
Using the same formula, the sum of the first \( pn \) terms \( S_{pn} \) is:
\[ S_{pn} = \frac{pn}{2} \left(2a + (pn-1)d\right) \]

3. **Form the Ratio \( \frac{S_{pn}}{S_n} \):**
Now, we compute the ratio:
\[ \frac{S_{pn}}{S_n} = \frac{\frac{pn}{2} \left(2a + (pn-1)d\right)}{\frac{n}{2} \left(2a + (n-1)d\right)} \]
Simplifying this expression gives:
\[ \frac{S_{pn}}{S_n} = \frac{p \left(2a + (pn-1)d\right)}{2a + (n-1)d} \]

4. **Condition for Independence of n:**
For \( \frac{S_{pn}}{S_n} \) to be independent of \( n \), the expression must not contain \( n \). Therefore, we analyze the terms:
\[ \frac{p \left(2a + (pn-1)d\right)}{2a + (n-1)d} \]
This implies that the coefficients of \( n \) in the numerator and denominator must cancel out.

5. **Setting Up the Equation:**
From the expression, we can equate the coefficients of \( n \):
- Coefficient of \( n \) in the numerator: \( pd \)
- Coefficient of \( n \) in the denominator: \( d \)

For independence from \( n \):
\[ pd = d \]
If \( d \neq 0 \), we can divide both sides by \( d \):
\[ p = 1 \]

6. **Finding the Relationship Between a and d:**
Now, substituting \( p = 1 \) back into the expression, we have:
\[ \frac{S_{n}}{S_{n}} = 1 \]
This does not provide new information. However, we also have:
\[ 2a + (pn-1)d = 2a + (n-1)d \]
This implies that:
\[ 2a = d \]
Therefore, we can express \( \frac{a}{d} \):
\[ \frac{a}{d} = \frac{a}{2a} = \frac{1}{2} \]

### Final Answer:
\[ \frac{a}{d} = \frac{1}{2} \]