Question

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

an=8,10,12,14,16, . . .
a subscript n baseline equals 8 comma 10 comma 12 comma 14 comma 16 comma dot dot dot

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, . .

Answer 1

To analyze the recursive definition given for the sequence:

1. The base term \( a_1 = 6 \).
2. The recursive formula \( a_n = a_{n-1} + 2 \).

This indicates that each term in the sequence is generated by taking the previous term and adding 2.

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

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

The sequence so far is \( 6, 8, 10, 12, 14, 16, \ldots \).

Now let's look at the provided response options:

1. **an = 8, 10, 12, 14, 16, ...** (This does not include 6 and does not match our full sequence.)
2. **an = 6 + 2n** (Let's check if this matches our sequence. For \( n = 1, a_1 = 6 + 2(1) = 8 \), which is incorrect.)
3. **an = 4 + 2n** (For \( n = 1, a_1 = 4 + 2(1) = 6 \), and for \( n = 2, a_2 = 4 + 2(2) = 8 \), which matches.)
4. **an = 6, 4, 2, 0, -2, ...** (This is a decreasing sequence and does not match.)

The correct option that represents the sequence defined by the recursion is:

**an = 4 + 2n** (Correct option)