The set `S_1 = {1}, S_2 = {3,5}, S_3 = {7,9,11}` , etc. forms a sequence.
Sum of all the elements of `S_(10)` is:

Options

A. 55
B. 300
C. 3375
D. none of these

Answer

To find the sum of all the elements of \( S_{10} \), we need to identify the elements in the set \( S_{10} \) based on the pattern established in the previous sets.

### Step-by-Step Solution:

1. **Identify the Pattern**:
- The sets \( S_n \) contain the first \( n \) odd numbers starting from 1.
- \( S_1 = \{1\} \) has 1 element.
- \( S_2 = \{3, 5\} \) has 2 elements.
- \( S_3 = \{7, 9, 11\} \) has 3 elements.
- Continuing this pattern, \( S_n \) contains \( n \) odd numbers.

2. **Determine the Elements of \( S_{10} \)**:
- Since \( S_{10} \) contains 10 odd numbers, we need to find the first 10 odd numbers starting from 1.
- The first 10 odd numbers are:
- 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

3. **Sum the Elements of \( S_{10} \)**:
- Now we will sum these numbers:
\[ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 \]

4. **Calculate the Sum**:
- We can pair the numbers for easier addition:
- (1 + 19) + (3 + 17) + (5 + 15) + (7 + 13) + (9 + 11)
- Each pair sums to 20:
\[ 20 + 20 + 20 + 20 + 20 = 100 \]

5. **Final Result**:
- Therefore, the sum of all the elements of \( S_{10} \) is \( 100 \).