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

Question

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.