A gentleman buys every year Bank's certificates of value exceeding the last year's purchase by Rs. 25. After 20 years he finds that the total value of the certificates purchased by him is Rs. 7250. Find the value of the certificates bought by him:
(i) in the first year
(ii) in the 13 th year.

Answer

To solve the problem, we can break it down into a series of steps:

### Step 1: Define the Variables
Let the value of the certificates bought in the first year be \( a \). According to the problem, the value of certificates bought each subsequent year increases by Rs. 25.

### Step 2: Write the Sequence
The values of the certificates bought over the years form an arithmetic progression (AP):
- First year: \( a \)
- Second year: \( a + 25 \)
- Third year: \( a + 50 \)
- ...
- 20th year: \( a + 475 \) (since \( 25 \times 19 = 475 \))

### Step 3: Total Value of Certificates
The total value of the certificates purchased over 20 years is given as Rs. 7250. The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic progression is:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Here, \( n = 20 \) and \( d = 25 \).

### Step 4: Substitute Values into the Formula
Substituting the known values into the formula:
\[ S_{20} = \frac{20}{2} \times (2a + (20-1) \times 25) \]
This simplifies to:
\[ 7250 = 10 \times (2a + 19 \times 25) \]
Calculating \( 19 \times 25 \):
\[ 19 \times 25 = 475 \]
So we have:
\[ 7250 = 10 \times (2a + 475) \]

### Step 5: Solve for \( a \)
Dividing both sides by 10:
\[ 725 = 2a + 475 \]
Subtracting 475 from both sides:
\[ 250 = 2a \]
Dividing by 2:
\[ a = 125 \]

### Step 6: Find the Value in the 13th Year
To find the value of the certificates bought in the 13th year, we use the formula for the \( n \)-th term of an arithmetic progression:
\[ A_n = a + (n-1)d \]
For the 13th year (\( n = 13 \)):
\[ A_{13} = 125 + (13-1) \times 25 \]
Calculating:
\[ A_{13} = 125 + 12 \times 25 \]
\[ 12 \times 25 = 300 \]
So:
\[ A_{13} = 125 + 300 = 425 \]

### Final Answers
(i) The value of the certificates bought in the first year is Rs. 125.
(ii) The value of the certificates bought in the 13th year is Rs. 425.

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