Which sequence has the given subscript notation a1=6

Question

; an=an−1+2
when defined recursively?(1 point)
Responses

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

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

Answer 1

To find which sequence matches the given recursive definition \( a_1 = 6 \) and \( a_n = a_{n-1} + 2 \), we can compute the first few terms of the sequence:

1. \( a_1 = 6 \)
2. \( a_2 = a_1 + 2 = 6 + 2 = 8 \)
3. \( a_3 = a_2 + 2 = 8 + 2 = 10 \)
4. \( a_4 = a_3 + 2 = 10 + 2 = 12 \)
5. \( a_5 = a_4 + 2 = 12 + 2 = 14 \)

Thus, the sequence generated by this recursive definition is: \[ a_n = 6, 8, 10, 12, 14, \ldots \]

Now let's check the options:

1. \( a_n = 6, 4, 2, 0, -2, \ldots \) - This does not match.
2. \( a_n = 4 + 2n \) - This does not represent the sequence generated as \( n \) increases.
3. \( a_n = 8, 10, 12, 14, 16, \ldots \) - This starts at 8, not 6, so it does not match.
4. \( a_n = 6 + 2n \) - This is indeed the correct form of the sequence.

If we set \( n = 1 \), \( a_1 = 6 + 2(1 - 1) = 6 \). If we set \( n = 2 \), \( a_2 = 6 + 2(2 - 1) = 8 \). If we set \( n = 3 \), \( a_3 = 6 + 2(3 - 1) = 10 \), and so on.

Thus, the correct choice that matches the given recursive definition is:

\( a_n = 6 + 2n \).