Which sequence has the given subscript notation a1=6

Question

; 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=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=6+2n

Answer 1

To determine which sequence corresponds to the given recursive notation \( a_1 = 6 \) and \( a_n = a_{n-1} + 2 \), we can analyze the sequence step by step.

1. Start with \( a_1 = 6 \).
2. For \( n = 2 \): \[ a_2 = a_1 + 2 = 6 + 2 = 8 \]
3. For \( n = 3 \): \[ a_3 = a_2 + 2 = 8 + 2 = 10 \]
4. For \( n = 4 \): \[ a_4 = a_3 + 2 = 10 + 2 = 12 \]
5. For \( n = 5 \): \[ a_5 = a_4 + 2 = 12 + 2 = 14 \]
6. For \( n = 6 \): \[ a_6 = a_5 + 2 = 14 + 2 = 16 \]

From these calculations, we can see that the sequential terms are: \( 6, 8, 10, 12, 14, 16, \ldots \).

Now, let's match this sequence to the provided options:

1. an=8,10,12,14,16, . . . - This sequence starts with 8 and does not match.
2. a subscript n baseline equals 4+2n - When \( n=1 \), \( a_1 = 4 + 2(1) = 6 \). This matches the first term. The formula gives \( a_n = 4 + 2n \) generates results \( 6, 8, 10, 12, \ldots \), so this is correct.
3. an=6,4,2,0,−2, . . . - This produces a decreasing sequence and does not match.
4. an=6+2n - For \( n=1 \), \( a_1 = 6 + 2(1) = 8 \). This does not match the first term.

Thus, the correct option is:

an = 4 + 2n (the second option).