A man borrows Rs. 8190 without interset and repays the loan in 12 monthly instalments. If each instalment is double the preceding one then the first and last instalments are (in rupees)

Options

A. 5 and 1200
B. 2 and 4096
C. 3 and 7200
D. none of these

Answer

To solve the problem, we need to find the first and last installments of a loan of Rs. 8190 that is repaid in 12 monthly installments, where each installment is double the preceding one.

### Step-by-Step Solution:

1. **Define the First Installment**:
Let the first installment be \( A \). The installments will then be:
- First installment (A1) = \( A \)
- Second installment (A2) = \( 2A \)
- Third installment (A3) = \( 4A \)
- Fourth installment (A4) = \( 8A \)
- Fifth installment (A5) = \( 16A \)
- Sixth installment (A6) = \( 32A \)
- Seventh installment (A7) = \( 64A \)
- Eighth installment (A8) = \( 128A \)
- Ninth installment (A9) = \( 256A \)
- Tenth installment (A10) = \( 512A \)
- Eleventh installment (A11) = \( 1024A \)
- Twelfth installment (A12) = \( 2048A \)

2. **Sum of All Installments**:
The total amount repaid is the sum of all installments:
\[ A + 2A + 4A + 8A + 16A + 32A + 64A + 128A + 256A + 512A + 1024A + 2048A = 8190 \]

3. **Factor Out A**:
We can factor out \( A \) from the left-hand side:
\[ A(1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048) = 8190 \]

4. **Calculate the Sum of the Series**:
The series \( 1 + 2 + 4 + 8 + ... + 2048 \) is a geometric series with the first term \( a = 1 \) and common ratio \( r = 2 \). The number of terms \( n = 12 \).
The sum of a geometric series can be calculated using the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S_{12} = 1 \cdot \frac{2^{12} - 1}{2 - 1} = 2^{12} - 1 = 4096 - 1 = 4095 \]

5. **Substituting Back**:
Now substituting back into the equation:
\[ A \cdot 4095 = 8190 \]

6. **Solve for A**:
\[ A = \frac{8190}{4095} = 2 \]

7. **Find the Last Installment**:
The last installment (A12) is:
\[ A_{12} = 2048A = 2048 \cdot 2 = 4096 \]

### Final Answer:
- The first installment is Rs. 2.
- The last installment is Rs. 4096.