The set `S_1 = {1}, S_2 = {3,5}, S_3 = {7,9,11}` , etc. forms a sequence.
The 11th element of the set `S_(21)` is :

Options

A. 21
B. 121
C. 221
D. 441

Answer

To find the 11th element of the set \( S_{21} \), we first need to understand the pattern in the sets \( S_n \).

1. **Identify the Pattern**:
- The first set \( S_1 = \{1\} \) has 1 element.
- The second set \( S_2 = \{3, 5\} \) has 2 elements.
- The third set \( S_3 = \{7, 9, 11\} \) has 3 elements.
- The fourth set \( S_4 = \{13, 15, 17, 19\} \) has 4 elements.
- The fifth set \( S_5 = \{21, 23, 25, 27, 29\} \) has 5 elements.

From this, we can see that the \( n \)-th set \( S_n \) contains \( n \) elements.

2. **Determine the Starting Point of Each Set**:
- The first element of \( S_1 \) is \( 1 \).
- The first element of \( S_2 \) is \( 3 \).
- The first element of \( S_3 \) is \( 7 \).
- The first element of \( S_4 \) is \( 13 \).
- The first element of \( S_5 \) is \( 21 \).

The first element of each set can be calculated as follows:
- The first element of \( S_n \) is given by the formula:
\[ \text{First element of } S_n = 1 + (n-1) \times 2 + (n-1)(n-2) \]
- This simplifies to \( 1 + (n-1)(n+1) \).

3. **Find the First Element of \( S_{21} \)**:
- For \( n = 21 \):
\[ \text{First element of } S_{21} = 1 + (21-1)(21+1) = 1 + 20 \times 22 = 1 + 440 = 441 \]

4. **List the Elements of \( S_{21} \)**:
- The elements of \( S_{21} \) will be:
\[ 441, 443, 445, \ldots \]
- The common difference is \( 2 \).

5. **Find the 11th Element**:
- The 11th element can be calculated using the formula for the \( n \)-th term of an arithmetic sequence:
\[ a_n = a + (n-1)d \]
- Here, \( a = 441 \), \( d = 2 \), and \( n = 11 \):
\[ a_{11} = 441 + (11-1) \times 2 = 441 + 20 = 461 \]

Thus, the 11th element of the set \( S_{21} \) is **461**.