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.
On which set the sum of all the elements of the set is even :

Options

A. a. `S_(39)`
B. b. `S_(50)`
C. c. `S_(72)`
D. d. `S_(94)`

Answer

To determine which sets \( S_n \) have an even sum of their elements, we first need to understand how the sets are constructed.

1. **Identify the pattern of the sets:**
- \( S_1 = \{1\} \) has 1 element.
- \( S_2 = \{2, 3\} \) has 2 elements.
- \( S_3 = \{4, 5, 6\} \) has 3 elements.
- \( S_4 = \{7, 8, 9, 10\} \) has 4 elements.
- \( S_5 = \{11, 12, 13, 14, 15\} \) has 5 elements.
- Continuing this pattern, \( S_n \) contains \( n \) elements.

2. **Determine the starting number of each set:**
- The first element of \( S_n \) can be calculated as follows:
- The first element of \( S_1 \) is 1.
- The first element of \( S_2 \) is 2.
- The first element of \( S_3 \) is 4.
- The first element of \( S_4 \) is 7.
- The first element of \( S_5 \) is 11.
- The first element of \( S_n \) can be derived from the formula:
\[ \text{First element of } S_n = \frac{n(n-1)}{2} + 1 \]
- This formula gives the sum of the first \( n-1 \) natural numbers plus 1.

3. **Calculate the sum of elements in each set:**
- The sum of the first \( n \) natural numbers starting from the first element can be calculated using:
\[ \text{Sum of } S_n = \text{First element} + \text{First element} + 1 + \text{First element} + 2 + ... + \text{First element} + (n-1) \]
- This can be simplified to:
\[ \text{Sum of } S_n = n \cdot \text{First element} + \frac{(n-1)n}{2} \]

4. **Check the parity of the sum:**
- We need to check for which \( n \) the sum is even.
- For \( n = 1, 2, 3, 4, 5, \ldots \):
- \( S_1: 1 \) (odd)
- \( S_2: 2 + 3 = 5 \) (odd)
- \( S_3: 4 + 5 + 6 = 15 \) (odd)
- \( S_4: 7 + 8 + 9 + 10 = 34 \) (even)
- \( S_5: 11 + 12 + 13 + 14 + 15 = 65 \) (odd)
- Continuing this way, we find that \( S_8 \) and \( S_{12} \) also yield even sums.

5. **Conclusion:**
- The sets \( S_n \) where the sum of all elements is even are those where \( n \) is divisible by 4 (i.e., \( S_4, S_8, S_{12}, \ldots \)).

### Final Answer:
The sets where the sum of all elements is even are \( S_4, S_8, S_{12}, \ldots \).