If `S_(1), S_(2), S_(3)` are the sums of n, 2n, 3n terms respectively of an A.P., then `S_(3)//(S_(2) - S_(1))-`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the value of \( \frac{S_3}{S_2 - S_1} \) where \( S_1, S_2, S_3 \) are the sums of \( n, 2n, 3n \) terms of an arithmetic progression (A.P.).

### Step-by-Step Solution:

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

2. **Calculate \( S_1 \)**:
For \( S_1 \) (sum of the first \( n \) terms):
\[ S_1 = \frac{n}{2} \left(2a + (n-1)d\right) \]

3. **Calculate \( S_2 \)**:
For \( S_2 \) (sum of the first \( 2n \) terms):
\[ S_2 = \frac{2n}{2} \left(2a + (2n-1)d\right) = n \left(2a + (2n-1)d\right) \]

4. **Calculate \( S_3 \)**:
For \( S_3 \) (sum of the first \( 3n \) terms):
\[ S_3 = \frac{3n}{2} \left(2a + (3n-1)d\right) \]

5. **Calculate \( S_2 - S_1 \)**:
Now, we need to find \( S_2 - S_1 \):
\[ S_2 - S_1 = n \left(2a + (2n-1)d\right) - \frac{n}{2} \left(2a + (n-1)d\right) \]
Simplifying this:
\[ S_2 - S_1 = n \left(2a + (2n-1)d\right) - \frac{n}{2} \left(2a + (n-1)d\right) \]
\[ = n \left(2a + 2nd - d\right) - \frac{n}{2} \left(2a + nd - d\right) \]
\[ = n(2a + 2nd - d) - \frac{n}{2}(2a + nd - d) \]
\[ = n \left(2a + 2nd - d - \frac{1}{2}(2a + nd - d)\right) \]
\[ = n \left(2a + 2nd - d - (a + \frac{nd}{2} - \frac{d}{2})\right) \]
\[ = n \left(2a - a + 2nd - \frac{nd}{2} - d + \frac{d}{2}\right) \]
\[ = n \left(a + \frac{4nd - nd}{2} - \frac{d}{2}\right) \]
\[ = n \left(a + \frac{3nd - d}{2}\right) \]

6. **Now Substitute \( S_3 \) and \( S_2 - S_1 \) into the Expression**:
Now we will calculate \( \frac{S_3}{S_2 - S_1} \):
\[ \frac{S_3}{S_2 - S_1} = \frac{\frac{3n}{2} \left(2a + (3n-1)d\right)}{n \left(a + \frac{3nd - d}{2}\right)} \]
Simplifying:
\[ = \frac{3}{2} \cdot \frac{2a + (3n-1)d}{a + \frac{3nd - d}{2}} \]

7. **Final Calculation**:
After simplification, we find that:
\[ \frac{S_3}{S_2 - S_1} = 3 \]

### Conclusion:
Thus, the value of \( \frac{S_3}{S_2 - S_1} \) is \( 3 \).