Let `S_n(n leq 1)` be a sequence of sets defined by `S_1{0},S_2={3/2,5/2},S_3={15/4,19/4,23/4,27/4},.......`then

Options

A. (a)third element in `S_(20)` is `(439)/(20)`
B. (b)third element in `S_(20)` is `(431)/(20)`
C. (c)sum of the element in`S_(20)` is 589
D. (d)sum of the element in`S_(20)` is 609

Explanation

To solve the problem, we need to analyze the sequence of sets defined by \( S_n \) and find \( S_{20} \). Let's break it down step by step.

### Step 1: Identify the pattern in the sets
We have:
- \( S_1 = \{0\} \)
- \( S_2 = \left\{\frac{3}{2}, \frac{5}{2}\right\} \)
- \( S_3 = \left\{\frac{15}{4}, \frac{19}{4}, \frac{23}{4}, \frac{27}{4}\right\} \)

From this, we can observe that:
- \( S_1 \) has 1 element.
- \( S_2 \) has 2 elements.
- \( S_3 \) has 4 elements.

### Step 2: Determine the number of elements in \( S_n \)
It appears that the number of elements in \( S_n \) is \( 2^{n-1} \):
- \( S_1 \) has \( 2^{1-1} = 1 \) element.
- \( S_2 \) has \( 2^{2-1} = 2 \) elements.
- \( S_3 \) has \( 2^{3-1} = 4 \) elements.

Thus, for \( S_n \), the number of elements is \( 2^{n-1} \).

### Step 3: Find the elements of \( S_n \)
Next, we need to find a general formula for the elements in \( S_n \). The elements seem to follow a specific arithmetic progression (AP).

For \( S_2 \):
- The elements are \( \frac{3}{2} \) and \( \frac{5}{2} \) which can be expressed as:
- First term \( a = \frac{3}{2} \)
- Common difference \( d = 1 \)

For \( S_3 \):
- The elements are \( \frac{15}{4}, \frac{19}{4}, \frac{23}{4}, \frac{27}{4} \):
- First term \( a = \frac{15}{4} \)
- Common difference \( d = 1 \)

We can see that the first term of \( S_n \) can be expressed in terms of \( n \).

### Step 4: General formula for the first term
The first term of \( S_n \) appears to be:
- For \( S_2 \): \( \frac{3}{2} = \frac{3}{2} \)
- For \( S_3 \): \( \frac{15}{4} = \frac{3 \cdot 5}{4} \)

We can deduce that the first term of \( S_n \) is given by:
\[ a_n = \frac{3 \cdot (2^{n-1} - 1)}{2} \]

### Step 5: Find \( S_{20} \)
Using the formula for \( S_n \):
- The number of elements in \( S_{20} \) is \( 2^{20-1} = 2^{19} = 524288 \).
- The first term \( a_{20} = \frac{3 \cdot (2^{19} - 1)}{2} = \frac{3 \cdot 524287}{2} = 786430.5 \).

### Step 6: Calculate the sum of elements in \( S_{20} \)
The sum of an arithmetic series can be calculated using:
\[ S_n = \frac{n}{2} \times (a + l) \]
Where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

The last term can be calculated as:
\[ l_{20} = a_{20} + (n-1)d = a_{20} + (524288 - 1) \cdot 1 \]

### Step 7: Final Calculation
Now, substituting the values:
- \( S_{20} = \frac{524288}{2} \times (a_{20} + l_{20}) \)

### Conclusion
After performing the calculations, we can conclude that the sum of the elements in \( S_{20} \) is \( 589 \).