The sequence of sets
`S_1,S_2,S_3,S_4,...` is defined as `S_1={1}, S_2={3,5}, S_3={7,9,11}, S_4={13,15,17,19}`… etc.
The first element of the nth set `S_n` is:

Options

A. n
B. `n_2+1`
C. 2n+1
D. `n^2-n+1`

Answer

To find the first element of the nth set \( S_n \) in the sequence defined as \( S_1 = \{1\}, S_2 = \{3, 5\}, S_3 = \{7, 9, 11\}, S_4 = \{13, 15, 17, 19\}, \ldots \), we will analyze the pattern of the first elements of each set.

### Step-by-Step Solution:

1. **Identify the first elements of the sets:**
- \( S_1 \): The first element is \( 1 \).
- \( S_2 \): The first element is \( 3 \).
- \( S_3 \): The first element is \( 7 \).
- \( S_4 \): The first element is \( 13 \).

2. **List the first elements and their corresponding set numbers:**
- For \( n = 1 \), the first element is \( 1 \).
- For \( n = 2 \), the first element is \( 3 \).
- For \( n = 3 \), the first element is \( 7 \).
- For \( n = 4 \), the first element is \( 13 \).

3. **Observe the pattern in the first elements:**
- The first elements are \( 1, 3, 7, 13 \).
- Let's find the differences between consecutive first elements:
- \( 3 - 1 = 2 \)
- \( 7 - 3 = 4 \)
- \( 13 - 7 = 6 \)

4. **Identify the pattern in the differences:**
- The differences are \( 2, 4, 6 \), which are consecutive even numbers.
- This suggests that the difference increases by \( 2 \) for each subsequent set.

5. **Formulate a general expression for the first element:**
- The first element of each set can be expressed as:
- For \( n = 1 \): \( 1 \)
- For \( n = 2 \): \( 1 + 2 = 3 \)
- For \( n = 3 \): \( 1 + 2 + 4 = 7 \)
- For \( n = 4 \): \( 1 + 2 + 4 + 6 = 13 \)

6. **Derive the formula:**
- The first element of the nth set can be expressed as:
\[ E_n = 1 + \sum_{k=1}^{n-1} (2k) \]
- This simplifies to:
\[ E_n = 1 + 2 \cdot \frac{(n-1)n}{2} = 1 + (n-1)n = n^2 - n + 1 \]

7. **Final expression for the first element of the nth set:**
- Therefore, the first element of the nth set \( S_n \) is:
\[ E_n = n^2 - n + 1 \]