For any natural number n the sets `S_1, S_2,…..` are defined as below:
`S_1 = {1} . S_2 = {2,3}, S_ 3 = {4,5,6}`
`S_4 = {7,8,9,10}, S_5 = {11,12,13,14,15}: ` etc.
The sum of the elements of set `S_(25)` is :

Options

A. 7825
B. 3125
C. 3250
D. none of these

Answer

To find the sum of the elements of the set \( S_{25} \), we first need to understand how the sets \( S_n \) are constructed.

1. **Identify the pattern in the sets**:
- \( S_1 = \{1\} \) has 1 element.
- \( S_2 = \{2, 3\} \) has 2 elements.
- \( S_3 = \{4, 5, 6\} \) has 3 elements.
- \( S_4 = \{7, 8, 9, 10\} \) has 4 elements.
- \( S_5 = \{11, 12, 13, 14, 15\} \) has 5 elements.

We can see that \( S_n \) contains \( n \) elements.

2. **Determine the starting number of each set**:
- The first set \( S_1 \) starts at 1.
- The second set \( S_2 \) starts at 2.
- The third set \( S_3 \) starts at 4.
- The fourth set \( S_4 \) starts at 7.
- The fifth set \( S_5 \) starts at 11.

To find the starting number of \( S_n \), we can observe that the starting number of \( S_n \) is the sum of the first \( n-1 \) natural numbers plus 1. This can be expressed as:
\[ \text{Starting number of } S_n = 1 + \sum_{k=1}^{n-1} k = 1 + \frac{(n-1)n}{2} \]

3. **Calculate the starting number of \( S_{25} \)**:
\[ \text{Starting number of } S_{25} = 1 + \frac{(25-1) \cdot 25}{2} = 1 + \frac{24 \cdot 25}{2} = 1 + 300 = 301 \]

4. **Determine the elements of \( S_{25} \)**:
Since \( S_{25} \) has 25 elements starting from 301, the elements of \( S_{25} \) are:
\[ S_{25} = \{301, 302, 303, \ldots, 325\} \]

5. **Calculate the sum of the elements in \( S_{25} \)**:
The sum of an arithmetic series can be calculated using the formula:
\[ \text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term}) \]
Here, \( n = 25 \), the first term is 301, and the last term is 325.
\[ \text{Sum} = \frac{25}{2} \times (301 + 325) = \frac{25}{2} \times 626 = 25 \times 313 = 7825 \]

Thus, the sum of the elements of set \( S_{25} \) is \( 7825 \).