A gentleman buys every year Banks' certificates of value exceeding the last year's purchase by 25. After 20 years he finds that the total value of the certificates purchased by him is 7,250. Find the value of the certificates purchased by him in the 1st year and in the 13th year.

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Define the Variables
Let the value of the certificates purchased in the first year be \( a \). Since the value of the certificates increases by 25 every year, the value of the certificates purchased in the second year will be \( a + 25 \), in the third year will be \( a + 50 \), and so on.

### Step 2: Identify the Sequence
The values of the certificates purchased over the years form an arithmetic progression (AP):
- 1st year: \( a \)
- 2nd year: \( a + 25 \)
- 3rd 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 7250. The formula for the sum \( S_n \) of the first \( n \) terms of an AP is:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( n = 20 \)
- \( d = 25 \)

Substituting the values into the formula:
\[ 7250 = \frac{20}{2} \times (2a + (20 - 1) \times 25) \]
\[ 7250 = 10 \times (2a + 475) \]

### Step 4: Simplify the Equation
Dividing both sides by 10:
\[ 725 = 2a + 475 \]
Now, isolate \( 2a \):
\[ 2a = 725 - 475 \]
\[ 2a = 250 \]
Now, divide by 2 to find \( a \):
\[ a = \frac{250}{2} = 125 \]

### Step 5: Value of Certificates in the 1st Year
Thus, the value of the certificates purchased in the first year is:
\[ \text{Value in 1st year} = a = 125 \]

### Step 6: Value of Certificates in the 13th Year
To find the value of the certificates purchased in the 13th year, we use the formula for the \( n \)-th term of an AP:
\[ a_n = a + (n - 1)d \]
Substituting \( n = 13 \):
\[ a_{13} = 125 + (13 - 1) \times 25 \]
\[ a_{13} = 125 + 12 \times 25 \]
\[ a_{13} = 125 + 300 = 425 \]

### Final Answers
- The value of the certificates purchased in the 1st year is **125**.
- The value of the certificates purchased in the 13th year is **425**.