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 analyze the recursive definition given for the sequence:

1. **Base case**: \( a_1 = 6 \)
2. **Recursive case**: \( a_n = a_{n-1} + 2 \)

We can calculate the first few terms of the sequence:

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

From this calculation, we can see that the sequence is: \( 6, 8, 10, 12, 14, \ldots \)

This represents an arithmetic sequence where the first term is 6 and the common difference is 2. The general formula for the \( n^{th} \) term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Where \( d \) is the common difference. In this case:

- \( a_1 = 6 \)
- \( d = 2 \)

So the formula becomes:

\[
a_n = 6 + (n - 1) \cdot 2 = 6 + 2n - 2 = 2n + 4
\]

This formula does not match the provided choices directly. However, analyzing the response options, we note that:

- \( a_n = 8, 10, 12, 14, 16, \ldots \) matches our calculated sequence, starting at 8.

Therefore, the correct option based on the recursive definition is:

**Response:**
- \( a_n = 8, 10, 12, 14, 16, \ldots \)