A person buys every year National Saving Certificates of value exceeding that last year's purchase by Rs. 500. After ten years he finds that the total value of the national saving certificates purchased by him is Rs. 27,500. Find the value of the certificates purchased by him in the first year.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Variables
Let the value of the National Saving Certificates purchased in the first year be \( X \).

### Step 2: Establish the Sequence
According to the problem, each year the value of the certificates purchased exceeds the previous year's purchase by Rs. 500. Therefore, the values of the certificates purchased over the years can be expressed as:
- First year: \( X \)
- Second year: \( X + 500 \)
- Third year: \( X + 1000 \)
- ...
- Tenth year: \( X + 4500 \)

### Step 3: Write the Total Value
The total value of the certificates purchased over 10 years is given as Rs. 27,500. We can express this total as the sum of the first 10 terms of an arithmetic progression (AP):
\[ \text{Total Value} = X + (X + 500) + (X + 1000) + \ldots + (X + 4500) \]

### Step 4: Use the Formula for the Sum of an AP
The sum \( S_n \) of the first \( n \) terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

Here, \( n = 10 \), \( a = X \), and \( d = 500 \).

### Step 5: Substitute Values into the Formula
Substituting the known values into the formula gives:
\[ S_{10} = \frac{10}{2} \times (2X + (10-1) \times 500) \]
\[ S_{10} = 5 \times (2X + 9 \times 500) \]
\[ S_{10} = 5 \times (2X + 4500) \]

### Step 6: Set the Equation Equal to the Total Value
We know that the total value is Rs. 27,500, so we set up the equation:
\[ 5 \times (2X + 4500) = 27500 \]

### Step 7: Solve for \( X \)
Dividing both sides by 5:
\[ 2X + 4500 = 5500 \]
Subtracting 4500 from both sides:
\[ 2X = 5500 - 4500 \]
\[ 2X = 1000 \]
Dividing by 2:
\[ X = 500 \]

### Conclusion
The value of the National Saving Certificates purchased in the first year is Rs. 500.

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