Question

Which sequence has the given subscript notation a1=6; an=an−1+2 when defined recursively?(1 point)
Responses

an=6+2n
a subscript n baseline equals 6 plus 2 n

an=4+2n
a subscript n baseline equals 4 plus 2 n

an=6,4,2,0,−2, . . .
a subscript n baseline equals 6 comma 4 comma 2 comma 0 comma negative 2 comma dot dot dot

an=8,10,12,14,16, . . .

Answer 1

To identify the correct form of the sequence defined by the notation \( a_1 = 6; \) and \( a_n = a_{n-1} + 2 \), we need to analyze the recursive definition.

1. **Initial term**: \( a_1 = 6 \)
2. **Recursion**: Each subsequent term can be found by adding 2 to the previous term.

Let's explore the first few terms of the sequence:

- \( a_1 = 6 \)
- \( a_2 = a_1 + 2 = 6 + 2 = 8 \)
- \( a_3 = a_2 + 2 = 8 + 2 = 10 \)
- \( a_4 = a_3 + 2 = 10 + 2 = 12 \)
- \( a_5 = a_4 + 2 = 12 + 2 = 14 \)
- \( a_6 = a_5 + 2 = 14 + 2 = 16 \)

The sequence continues as:
\[ 6, 8, 10, 12, 14, 16, \ldots \]

This matches the option:
- \( a_n = 8, 10, 12, 14, 16, \ldots \)

However, none of the options correctly starts with 6. Still, let's analyze the options provided:

1. **\( a_n = 6 + 2n \)**: This is not correct because it does not match with the initial term and the sequence derived.
2. **\( a_n = 4 + 2n \)**: This will yield a different starting value as it would start from 4, which is not correct.
3. **\( a_n = 6, 4, 2, 0, -2, \ldots \)**: This is a decreasing sequence and does not match our derived sequence.
4. **\( a_n = 8, 10, 12, 14, 16, \ldots \)**: The correct pattern \( a_2, a_3, \ldots \) but not including \( a_1 = 6 \).

Given that none of the options perfectly represent the sequence defined, but the closest logical sequence that starts from the second term is:

**The best choice based on the pattern shown is:**
- **\( a_n = 8, 10, 12, 14, 16, \ldots \)**.

However, please note that this does not reflect the entire recursion accurately as it begins from \( a_2 \). The true series starts with 6 and increases by 2 for further terms as shown.