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 last element in the `S_(24)` is :

Options

A. 599
B. 600
C. 300
D. 625

Answer

To find the last element in the set \( S_{24} \), we first need to understand the pattern in the sets defined as \( S_1, S_2, \ldots \).

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.

From this, we can see that \( S_n \) contains \( n \) elements.

2. **Calculate the total number of elements up to \( S_{23} \)**:
- The total number of elements in the first \( n \) sets can be calculated using the formula for the sum of the first \( n \) natural numbers:
\[ \text{Total elements} = 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
- For \( n = 23 \):
\[ \text{Total elements in } S_{23} = \frac{23 \times (23 + 1)}{2} = \frac{23 \times 24}{2} = 276 \]

3. **Determine the first element of \( S_{24} \)**:
- The first element of \( S_{24} \) will be the next number after the last element of \( S_{23} \).
- Since \( S_{23} \) has 276 elements, the last element of \( S_{23} \) is 276.
- Therefore, the first element of \( S_{24} \) is \( 276 + 1 = 277 \).

4. **Find the last element of \( S_{24} \)**:
- \( S_{24} \) contains 24 elements, starting from 277.
- The last element of \( S_{24} \) can be calculated as:
\[ \text{Last element of } S_{24} = 277 + (24 - 1) = 277 + 23 = 300 \]

Thus, the last element in the set \( S_{24} \) is **300**.