The set `S_1={1}` `S_2={3,5}`, `S_3`={7,9,11} etc. forms a sequence.
The sum of the first and last element of the set `S_51` is :

Options

A. 5202
B. 5151
C. 5152
D. 5102

Answer

To solve the problem, we need to find the first and last elements of the set \( S_{51} \) and then calculate their sum.

### Step-by-Step Solution:

1. **Identify the pattern in the sets:**
- The set \( S_1 \) has 1 element: \( \{1\} \).
- The set \( S_2 \) has 2 elements: \( \{3, 5\} \).
- The set \( S_3 \) has 3 elements: \( \{7, 9, 11\} \).
- Continuing this pattern, \( S_n \) has \( n \) elements, which are the next \( n \) odd numbers.

2. **Find the first element of \( S_{51} \):**
- The first element of \( S_n \) can be determined using the formula:
\[ \text{First element of } S_n = n^2 - n + 1 \]
- For \( n = 51 \):
\[ \text{First element of } S_{51} = 51^2 - 51 + 1 \]
\[ = 2601 - 51 + 1 \]
\[ = 2601 - 50 = 2551 \]

3. **Find the last element of \( S_{51} \):**
- The last element of \( S_n \) is the odd number before the first element of \( S_{n+1} \).
- First, we find the first element of \( S_{52} \):
\[ \text{First element of } S_{52} = 52^2 - 52 + 1 \]
\[ = 2704 - 52 + 1 \]
\[ = 2704 - 51 = 2653 \]
- The last element of \( S_{51} \) is the odd number just before 2653, which is:
\[ 2653 - 2 = 2651 \]

4. **Calculate the sum of the first and last elements of \( S_{51} \):**
- Now we have:
- First element: \( 2551 \)
- Last element: \( 2651 \)
- The sum is:
\[ 2551 + 2651 = 5202 \]

### Final Answer:
The sum of the first and last element of the set \( S_{51} \) is \( 5202 \).