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 middlemost element of the set `S_(15)` is:

Options

A. 196
B. 169
C. 131
D. none of these

Answer

To find the middlemost element of the set \( S_{15} \), we first need to determine the elements of the set \( S_{15} \).

### Step 1: Determine the pattern of the sets
The sets \( S_n \) are defined such that:
- \( S_1 \) has 1 element
- \( S_2 \) has 2 elements
- \( S_3 \) has 3 elements
- \( S_4 \) has 4 elements
- ...
- \( S_n \) has \( n \) elements

### Step 2: Calculate the total number of elements in the first 14 sets
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:
\[ \text{Total elements in } S_1, S_2, \ldots, S_n = \frac{n(n + 1)}{2} \]
For \( n = 14 \):
\[ \text{Total elements in } S_1, S_2, \ldots, S_{14} = \frac{14 \times 15}{2} = 105 \]

### Step 3: Determine the starting element of \( S_{15} \)
The set \( S_{15} \) will start from the next number after the last element of \( S_{14} \). The last element of \( S_{14} \) can be found as follows:
- The last element of \( S_{14} \) is the 105th element.
- Since \( S_{14} \) has 14 elements, the last element is:
\[ 105 + 1 = 106 \]
Thus, \( S_{15} \) starts from 106.

### Step 4: Determine the elements of \( S_{15} \)
The set \( S_{15} \) will have 15 elements starting from 106:
\[ S_{15} = \{106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120\} \]

### Step 5: Find the middlemost element
To find the middlemost element in \( S_{15} \):
- The number of elements in \( S_{15} \) is 15, which is odd.
- The middlemost element can be found at position \( \frac{15 + 1}{2} = 8 \).

### Step 6: Identify the 8th element
The 8th element in the set \( S_{15} \) is:
\[ S_{15}[8] = 113 \]

### Conclusion
The middlemost element of the set \( S_{15} \) is **113**.

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