The sequence of sets `S_1, S_2,S_3,S_4,…..` is defined as `S_1 = {1}. S_2 = {3,5}, S_3 = {7,9,11}, S_4 = {13,15,17,19}`….etc.
The middlemost element of an odd numbered set `S_(125)` is:

Options

A. 12500
B. 15625
C. 3125
D. none of these

Answer

To find the middlemost element of the odd-numbered set \( S_{125} \), we can follow these steps:

### Step 1: Identify the pattern of the first element of each set
The first elements of the sets are as follows:
- \( S_1 = \{1\} \) → First element is \( 1 \)
- \( S_2 = \{3, 5\} \) → First element is \( 3 \)
- \( S_3 = \{7, 9, 11\} \) → First element is \( 7 \)
- \( S_4 = \{13, 15, 17, 19\} \) → First element is \( 13 \)

We can see that the first elements of the sets form a sequence where the difference between consecutive first elements increases by \( 2 \) for each subsequent set.

### Step 2: Determine the first element of \( S_n \)
The first element of each set can be derived from the formula:
- For \( S_n \), the first element can be calculated as:
\[ \text{First element of } S_n = n^2 - n + 1 \]

### Step 3: Calculate the first element of \( S_{125} \)
Substituting \( n = 125 \) into the formula:
\[ \text{First element of } S_{125} = 125^2 - 125 + 1 \]
Calculating \( 125^2 \):
\[ 125^2 = 15625 \]
Now substituting back:
\[ \text{First element of } S_{125} = 15625 - 125 + 1 = 15501 \]

### Step 4: Determine the number of elements in \( S_{125} \)
The number of elements in \( S_n \) is equal to \( n \). Therefore, \( S_{125} \) has \( 125 \) elements.

### Step 5: Find the middlemost element
Since \( S_{125} \) has \( 125 \) elements (which is odd), the middlemost element is the \( 63^{rd} \) element (since \( \frac{125 + 1}{2} = 63 \)).

### Step 6: Calculate the \( 63^{rd} \) element of \( S_{125} \)
The elements of \( S_{125} \) are an arithmetic sequence starting from \( 15501 \) with a common difference of \( 2 \). The \( n^{th} \) term of an arithmetic sequence can be calculated using the formula:
\[ T_n = a + (n - 1) \cdot d \]
Where:
- \( a = 15501 \) (first term)
- \( d = 2 \) (common difference)
- \( n = 63 \)

Substituting the values:
\[ T_{63} = 15501 + (63 - 1) \cdot 2 \]
Calculating:
\[ T_{63} = 15501 + 62 \cdot 2 = 15501 + 124 = 15625 \]

### Final Answer
The middlemost element of the set \( S_{125} \) is \( 15625 \).

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