A sum of रु6240 is paid off in 30 instalments, such that each instalment is 10 more than the preceding instalment. Calculate the value of the first instalment.

Answer

To solve the problem step by step, we need to find the value of the first installment given that a total of ₹6240 is paid in 30 installments, where each installment is ₹10 more than the preceding one.

### Step-by-step Solution:

1. **Define Variables:**
Let the first installment be \( a \). Then, the second installment will be \( a + 10 \), the third installment will be \( a + 20 \), and so on.

2. **Identify the Series:**
The installments form an arithmetic progression (AP) where:
- First term \( a \)
- Common difference \( d = 10 \)
- Number of terms \( n = 30 \)

3. **Use the Formula for the Sum of an AP:**
The sum \( S_n \) of the first \( n \) terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Here, \( S_n = 6240 \), \( n = 30 \), and \( d = 10 \).

4. **Substitute Values into the Formula:**
Substitute \( S_n \), \( n \), and \( d \) into the formula:
\[ 6240 = \frac{30}{2} \times (2a + (30-1) \times 10) \]
Simplifying this gives:
\[ 6240 = 15 \times (2a + 290) \]

5. **Simplify the Equation:**
Divide both sides by 15:
\[ 416 = 2a + 290 \]

6. **Isolate \( 2a \):**
Subtract 290 from both sides:
\[ 2a = 416 - 290 \]
\[ 2a = 126 \]

7. **Solve for \( a \):**
Divide by 2 to find \( a \):
\[ a = \frac{126}{2} = 63 \]

8. **Conclusion:**
Therefore, the value of the first installment is \( \text{₹} 63 \).